STUDY OF FUZZY MEASURE AND SOME PROPERTIES OF NULL-ADDITIVE FUZZY MEASUREThesisSubmitted to theKUMAUN UNIVERSITY, NAINITAL, Uttarakhand, INDIA

IN PARTIAL FULFILMENT OF THE REQUIREMENTSFOR THE DEGREE OFDoctor of Philosophy(Mathematics)May, 2015UNDER THE SUPERVISION OFDR. H.S. NAYALAssociate ProfessorDepartment of mathematicsGovt. Postgraduate College, Ranikhet (Almora)UTTARAKHAND.

SUBMITTED BYPARUL AGARWALM. Sc., M. Phil. (Mathematics)

2015ACKNOWLEDGMENTSThe precious gift of learning is a debt that is difficult to pay. Only gratitude can be left. Indeed, the words at my command are inadequate in form or in sprit to express my heartfelt, deep sense of unbound gratitude and indebtedness to Dr. H. S. Nayal, Associate Professor, Department of mathematics, Govt. Postgraduate College, Ranikhet (Almora), Uttarakhand. I feel golden opportunity and proud privilege to work under the inspiring guidance, imperative and facial suggestions of Dr. H. S. Nayal, who is not just a guide, but an angelic to me. He has not only helped me in my research work but also provided benevolent guidance during the whole degree programme. I have real appreciation and regard for him for his magnanimous and cooperative attitude, painstaking efforts, peerless criticism and especially for realizing to capitalize on my strength to work hard and acquire confidence to deal with every difficulty.

I am immensely grateful to Dr. Sushil Jain and Dr. Priyanka Pathak, Department of education, Govt. Postgraduate College, Ranikhet (Almora), Uttarakhand for their scholastic guidance, inspiring suggestions and help at various stages of the research work and for their all time support, blessing, impeccable counsel and cooperation during the course of present study.

I have deepest respect and regards for my father, Shri R. N. Agarwal and mother, Smt. Kusum Agarwal for their encouragement, motivation and valuable guidance during this academic venture. I dont have words to express my gratitude and indebtedness to my brother, Mr. Saurabh Agarwal, who has always motivated and inspired me to work. From the deepest core of my heart, I express very affectionate thanks to my sister, Dr. Nitu Agarwal, whose emotive support and care enabled me to bring this work in shape. The love, affection and encouragement that I got from my family are unforgettable. They are my strength, my support and my pride.

I find myself incapable to express my warmest thanks to my seniors, juniors, and friends for their encouragement, suggestions, moral support during my study period and for the enjoyable and precious moments shared with them.Research fellowship provided by Kumaun University, Nainital during period of my doctoral degree programme is thankfully acknowledged.

Ranikhet (Parul Agarwal)April,2015 Authoress

ContentsCHAPTER 1Introduction

1.1 Measure

1.2 Fuzzy

1.3 Uncertainty

1.4 Fuzzy measure

1.5 Null-additive fuzzy measure

CHAPTER 2Possibility Theory versus Probability Theory in Fuzzy Measure Theory2.1 Concept of associative probabilities to a fuzzy measure

2.2 Shannon Entropy of fuzzy measure

2.3 Basic properties of entropy of fuzzy measure

2.4 Application of entropy of fuzzy measure

CHAPTER 3Properties of Null-Additive and Absolute Continuity of a Fuzzy Measure3.1 Absolute continuity of a fuzzy measure

3.2 Some results on null-additive fuzzy measure

3.3 Lebesgue decomposition type null-additive fuzzy measure

3.4 Generalization of the symmetric fuzzy measure

3.5 Properties of fuzzy measures

CHAPTER 4Properties of Strong Regularity of Fuzzy Measure on Metric Space4.1 Null additive fuzzy measure on metric space

4.2 Strong regularity of fuzzy measure on metric space

4.3 Properties of inner\outer regularity of fuzzy measure

CHAPTER 5Continuous, auto-continuous and completeness of fuzzy measure5.1 Continuous and auto-continuous of fuzzy measure

5.2 Results on completeness of fuzzy measure space

5.3 Convergence in fuzzy measure

5.4 Some other properties of fuzzy measure and convergence

BIBLIOGRAPHY

Introduction1.1 Measure:In mathematical analysis, a measure on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size by [29]. In this sense, a measure is a generalization of the concept of length, area, and volume. A particularly important example is the Lebesgue measure on a Euclidean space, which assigns the conventional length, area, and volume of Euclidean geometry to suitable subsets of the n-dimensional Euclidean space , according to [29], [86].Measure theory was developed in successive stage during the late 19th and early 20th centuries by mile Borel, Henri Lebesgue, Johann Radon and Maurice Frchet, among others, by [100], [18], [31]. The main applications of measures are in the foundations of the Lebesgue integral [64], in Andrey Kolmogorovs axiomatisation of probability theory and in ergodic theory [42]. In integration theory, specifying a measure allows one to define integrals on spaces more general than subsets of Euclidean space; moreover, the integral with respect to the Lebesgue measure on Euclidean spaces is more general and has a richer theory than its predecessor, the Riemann integral (see [29]). Probability theory considers measures that assign to the whole set of size 1, and considers measurable subsets to be events whose probability is given by the measure by [86]. Ergodic theory considers measures that are invariant under, or arise naturally from, a dynamical system [42]. Technically, a measure is a function that assigns a non-negative real number or to certain subsets of a set X [42]. It must assign 0 to the empty set and be countably additive: the measure of a large subset that can be decomposed into a countable number of smaller disjoint subset, is the sum of the measures of the smaller subsets by [9]. In general, if one wants to associate a consistent size to each subset of a given set while satisfying the other axioms of a measure, one only finds trivial examples like the counting measure [100]. This problem was resolved by defining measure only on a sub-collection of all subsets; the so called measurable subsets, which are required to form a algebra [18], this means that countable unions, countable intersections and complements of measurable subsets are measurable. 1.1.1Definition:Let X be a set and a algebra over X. A function from to the extended real number line i.e. : is called a measure if it satisfies the following properties: according to [79].(a) Non-negativity: for all (b) Null empty set:(c) Countable additivity (or - additivity):For all countable collections of pairwise disjoint sets in :

One may require that at least one set E has finite measure. Then the null set automatically has measure zero because of countable additivity, because

is finite if and only if the empty set has measure zero. If only the second and third conditions of the definition of measure above are met, and takes on at most one of the values , then is called a signed measure by [29], [18].1.1.2Properties:Several further properties can be derived from the definition of a countably additive measure by [31], [9], [79].(a) Monotonicity: A measure is monotonic: If and are measurable sets with then, . (1c)(b) Measures of infinite unions of measurable sets:A measure is countably subadditive: If E1, E2, E3, . is a countable sequence of sets in , not necessarily disjoint, then

A measure is continuous from below: If E1, E2, E3, . are measurable sets and En is a subset of En+1 for all n, then the union of the sets En is measurable, and

(d) Measures of infinite intersections of measurable sets:A measure is continuous from above: If E1, E2, E3, . are measurable sets and En+1 is a subset of En for all n, then the intersection of the sets En is measurable; furthermore, if at least one of the En has finite measure, then

This property is false without the assumption that at least one of the En has finite measure. For instance, for each , let , which all have infinite Lebesgue measure, but the intersection is empty.1.1.3.Some measures are defined here by [9], [18], [79], [100]ISigma-finite measure: A measure space (X, , ) is called finite if (X) is a finite real number. It is called - finite if X can be decomposed into a countable union of measurable sets of finite measure. A set in a measure space has finite measure if it is a countable union of sets with finite measure.For example, the real numbers with the standard Lebesgue measure are finite but not finite. Consider the closed intervals [k, k+1] for all integers k; there are countably many such intervals, each has measure 1, and their union is the entire real line. Alternatively, consider the real numbers with the counting measure, which assigns to each finite set of real numbers i.e. number of points in the set. This measure space is not finite, because every set with finite measure contains only finitely many points, and it would take uncountably many such set to cover the entire real line. The finite measure spaces have some very convenient properties; finiteness can be compared in this respect to the Lindelf property of topological spaces. They can be also thought of as a vague generalization of the idea that a measure space may have uncountable measure.IIComplete measure: A measurable set X is called a null set if (X). A subset of a null set is called a negligible set. A negligible set need not be measurable, but every measurable negligible set is automatically a null set. A measure is called complete measure if every negligible set is measurable.A measure can be extended to a complete one by considering the algebra of subsets Y which differ by a negligible set from a measurable set X, that is, such that the symmetric difference of X and Y is contained in a null set.III - Additive measure: A measure on is additive if for any and any family the following conditions hold;

Note that the second condition is equivalent to the statement that the ideal of null sets is - complete. Some other important measures are listed here by [29], [86], [9].IVLebesgue Measure: The Lebesgue measure on R is a complete translation-invariant measure on a -algebra containing the intervals in R such that ; and every other measure with these properties extends Lebesgue measure.The Hausdorff measure is a generalization of the Lebesgue measure to sets with non-integer dimension, in particular, fractal sets. The Haar measure for a locally compact topological group is also a generalization of the Lebesgue measure (and also of counting measure and circular angle measure) and has similar uniqueness properties. Circular angle measure is invariant under rotation, and hyperbolic angle measure is invariant under squeeze mapping.Other measures used in various theories include: Borel measure, Jordan measure, Ergodic measure, Euler measure, Gaussian measure, Baire measure, Radon measure, Young measure and so on by [31], [79].1.1.4. Generalizations: For certain purposes, it is useful to have a measure whose values are not restricted to the non-negative real or infinity. For instance, a countably additive set function with values in the (signed) real numbers is called a signed measure, while such a function with values in the complex numbers is called a complex measure. Measures that take values in Banach spaces have been studied extensively by [42]. A measure that takes values in the set of self-adjoint projections on a Hilbert space is called a projection-valued measure; these are used in functional analysis for the spectral theorem. When it is necessary to distinguish the usual measure, which take non-negative values from generalizations, the term positive measure is used. Positive measures are closed under conical combination but not general linear combination, while signed measures are the linear closure of positive measures [100].Another generalization is the finitely additive measure, which are sometimes called contents. This is the same as a measure except that instead of requiring countable additivity we require only finite additivity. Finite additive measures are connected with notions such as Banach limits, the dual of and the Stone-Cech compactification by [9]. All these are linked in one way or another to the axiom of choice.1.2.Fuzzy:Fuzzy mathematics forms a branch of mathematics related to fuzzy set theory and fuzzy logic. It started in 1965 after the publication of Lotfi Asker Zadehs seminal work Fuzzy Sets by [60]. A fuzzy subset A of a set X is a function , where L is the interval [0,1]. This function is also called a membership function. A membership function is a generalization of a characteristic function or an indicator function of a subset defined for . More generally, one can use a complete lattice L in a definition of a fuzzy subset A [55].The evolution of the fuzzification of mathematical concepts can be broken down into three stages: by [23](a) straightforward fuzzification during the sixties and seventies,(b) the explosion of the possible choices in the generalization process during the eighties,(c) the standardization, axiomatization and L-fuzzification in the nineties.Usually, a fuzzification of mathematical concepts is based on a generalization of these concepts from characteristic functions to membership functions. Let A and B be two fuzzy subsets of X. Intersection and union are defined as follows:

Instead of min and max one can use t-norm and t-conorm, respectively, by [24] for example, min (a, b) can be replaced by multiplication ab. A straightforward fuzzification is usually based on min and max operations because in this case more properties of traditional mathematics can be extended to the fuzzy case.A very important generalization principle used in fuzzification of algebraic operations is a closure property. Let * be a binary operation on X. The closure property for fuzzy subsets A of X is that for all Let (G, *) be a group and A a fuzzy subset of G. Then A is a fuzzy subgroup of G if for all x, y in G, [24].A fuzzy is a concept of which the meaningful content, value, or boundaries of application can vary considerably according to context or conditions, instead of being fixed once and for all [97]. This generally means the concept is vague, lacking a fixed, precise meaning, without however being meaningless altogether [91]. It has a meaning, or multiple meaning. But these can become clearer only through further elaboration and specification, including a closer definition of the context in which they are used. Fuzzy concepts lack clarity and are difficult to test or operationalize [6].In logic, fuzzy concepts are often regarded as concepts which in their application, or formally speaking, are neither completely true nor completely false, or which are partly true or partly false; they are ideas which require further elaboration, specification or qualification to understand their applicability [75] (the conditions under which they truly make sense).In mathematics and statistics, a fuzzy variable (such as the temperature, hot or cold) is a value which could lie in a probable range defined by quantitative limits or parameters, and which can be usefully described with imprecise categories (such as high, medium of low).In mathematics and computer science, the gradations of applicable meaning of a fuzzy concept are described in terms of quantitative relationships defined by logical operators. Such an approach is sometimes called degree-theoretic semantics by logicians and philosophers [92], but more usual term is fuzzy logic or many-valued logic. The basic idea is, that a real number is assigned to each statement written in a language, within a range from 0 to 1, where 1 means that the statement is completely true, and 0 means that the statement is completely false, while values less than 1 but greater than 0 represent that the statements are partly true, to a given, quantifiable extent. This makes it possible to analyze a distribution of statement for their truth-content, identify data patterns, make inferences and predictions, and model how processes operate.Fuzzy reasoning (i.e. reasoning with graded concepts) has many practical uses [57]. It is nowadays widely used in the programming of vehicle and transport electronics, household appliances, video games, language filters, robotics, and various kinds of electronic equipment used for pattern recognition, surveying and monitoring(such as radars). Fuzzy reasoning is also used in artificial intelligence and virtual intelligence research [74] Fuzzy risk scores are used by project managers and portfolio managers to express risk assessments [49].A fuzzy set can be defined mathematically by assigning to each possible individual in the universe of discourse a value representing its grade of membership in the fuzzy set. This grade corresponds to the degree to which that individual is similar or compatible with the concept represented by the fuzzy set. Thus, individuals may belong in the fuzzy set to a greater or lesser degree as indicated by a larger or smaller membership grade. As already mentioned, these membership grades are very often represented by real-number values ranging in the closed interval between 0 and 1(by [37] on page 4-5). Thus, a fuzzy set representing our concept of sunny might assign a degree of membership of 1 to a cloud cover of 0%, 0.8 to a cloud cover of 20%, 0.4 to a cloud cover of 30%, and 0 to a cloud cover of 75%. These grades signify the degree to which each percentage of cloud cover approximate our subjective concept of sunny, and the set itself models the semantic flexibility inherent in such a common linguistic term. Because full membership and full non-membership in the fuzzy set can still be indicated by the values of 1 and 0, respectively, we can consider the concept of a crisp set to be a restricted case of the more general concept of a fuzzy set for which only these two grades of membership are allowed [37]. 1.3.Uncertainty:Fuzzy concepts can generate uncertainty because they are imprecise (especially if they refer to a process in motion or a process of transformation where something is in the process of turning into something else). In that case, they do not provide a clear orientation for action or decision-making, reducing fuzziness, perhaps by applying fuzzy logic, would generate more certainty [75]. A fuzzy concept may indeed provide more security, because it provides a meaning for something when an exact concept is unavailable which is better than not being able to denote it at all [90].It is generally agreed that an important point in the evolution of the modern concept of uncertainty was the publication of a seminal paper by [60], even though some ideas presented in the paper were envisioned some few years earlier by [76]. In this paper, Zadeh introduced a theory whose objects fuzzy sets are sets with boundaries that are not precise. The membership in a fuzzy set is not a matter of affirmation or denial, but rather a matter of a degree [37].The significance of [60] was that it challenged not only probability theory as the sole agent for uncertainty, but the very foundations upon which probability theory is based: Aristotelian two-valued logic. When A is a fuzzy set and x is a relevant object, the proposition x is a member of A is not necessarily either true or false, as required by two-valued logic, but it may be true only to some degree, the degree to which x is actually a member of A. it is most common, but not required, to express degrees of membership in fuzzy sets as well as degree of truth of the associated propositions by numbers in the closed unit interval [0,1]. The extreme values in this interval, 0 and 1, then represent, respectively, the total denial and affirmation of the membership in a given fuzzy set as well as the falsity and truth of the associated proposition.For instance, suppose we are trying to diagnose an ill patient. In simplified terms, we may be trying to determine whether this patient belongs to the set of people with, say, pneumonia, bronchitis, emphysema, or a common cold. A physical examination may provide us with helpful yet inconclusive evidence (by [37] on page 179). For example, we might assign a high values, say 0.75, to our best guess, bronchitis, and a lower value to the other possibilities, such as 0.45 for the set consisting of pneumonia and emphysema and 0 for a common cold. These values reflect the degrees to which the patients symptoms provide evidence for the individual diseases or set of diseases; their collection constitutes a fuzzy measure representing the uncertainty associated with several well-defined alternatives. It is important to realize that this type of uncertainty, which results from information deficiency, is fundamentally different from fuzziness, which results from the lack of sharp boundaries.Uncertainty-based information was first conceived in terms of classical set theory and in terms of probability theory. The term information theory has almost invariably been used to a theory based upon the well known measure of probabilistic uncertainty established by [12]. Research on a broader conception of uncertainty-based information, liberated from the confines of classical set theory and probability theory, began in the early eighties. The name generalized information theory was coined for a theory based upon this broader conception.The ultimate goal of generalized information theory is to capture properties of uncertainty-based information formalized within any feasible mathematical framework. Although this goal has not been fully achieved as yet, substantial progress has been made in this direction. In addition to classical set theory and probability theory, uncertainty-based information is now well understood in fuzzy set theory, possibility theory and evidence theory.Three types of uncertainty are now recognized in the five theories, in which measurement of uncertainty is currently well established. These three uncertainty types are: nonspecificity (or imprecision), which is connected with sizes (cardinalities) of relevant sets of alternatives; fuzziness (or vagueness), which results from imprecise boundaries of fuzzy sets; and strife (or discord), which expresses conflicts among the various sets of alternatives.1.3.1Nonspecificity (or Imprecision): Measurement of uncertainty and associated information was first conceived in terms of classical set theory. It was shown by [39] that using a function from the class of functions(1g)where denotes the cardinality of a finite nonempty set A, and b, c are positive constants , is the only sensible way to measure the amount of uncertainty associated with a finite set of possible alternatives. Each choice of values of the constants b and c determines the unit in which uncertainty is measured. When b=2 and c=1, which is most common choice, uncertainty is measured in bits, and obtain (1h)One bit of uncertainty is equivalent to the total uncertainty regarding the truth or falsity of one proposition. The set function U defined by equation (1h) be called a Hartley function. Its uniqueness as a measure of uncertainty associated with sets of alternatives can also be proven axiomatically.A natural generalization of the Hartley function from classical set theory to fuzzy set theory was proposed in the early eighties under the name U- uncertainty. For any nonempty fuzzy set A defined on a finite universal set X, the generalized Hartley function [39] has the form

where denotes the cardinality of the - cut of A and h(A) is the height of A. Observe that U(A), which measures nonspecificity of A, is a weighted average of values of the Hartley function for all distinct - cuts of the normalized counterpart of A, defined by A(x)/h(A) for all xX. Each weight is a difference between the values of of a given -cut and the immediately preceding -cut. Fuzzy sets are equal when normalized have the same nonspecificity measured by function U [39].

1.3.2.Fuzziness (or Vagueness): The second type of uncertainty that involves fuzzy sets (but not crisp sets) is fuzziness (or vagueness). In general, a measure of fuzziness is a function where P(X) denotes the set of all fuzzy subsets of X (fuzzy power set). For each fuzzy set A, this function assign a nonnegative real number f(A) that express the degree to which the boundary of A is not sharp [37]. One way is to measure fuzziness of any set A by a metric distance between its membership grade function and the membership grade function (or characteristic function) of the nearest crisp set [60]. Even when committing to this conception of measuring fuzziness, the measurement is not unique. To make it unique, we have to choose a suitable distance function.Another way of measuring fuzziness, which seems more practical as well as more general, is to view the fuzziness of a set in terms of the lack of distinction between the set and it complement. Indeed, it is precisely the lack of distinction between the sets and their complements that distinguishes fuzzy sets from crisp sets [60]. The less a set differs from its complement, the fuzzier it is. Let us restrict our discussion to this view of fuzziness, which is currently predominant in the literature.1.3.3.Strife (or discord): This type of uncertainty which is connected with conflicts among evidential claims has been far more controversial. Although it is generally agreed that a measure of this type of uncertainty must be a generalization of the well-established Shannon entropy [39] from probability theory, which all collapse to the Shannon entropy within the domain of probability theory, is the right generalization. As argued in the companion paper [35], either of these measures is deficient in some trails. To overcome these deficiencies, another measure was prepared in the companion paper [35], which is called a measure of discord. This measure is expressed by a function D and defined by the formula,

The rationale for choosing this function is explained as follows. The term,

In equation (1j) expresses the sum of conflicts of individual evidential claim m(B) for all with respect to the evidential claim m(A) focusing on a particular set A; each individual conflict is properly scaled by the degree to which the subsethood relation is violated. The function , which is employed in equation (1j), is monotonic increasing with Con (A) and, consequently, it represents the same quantity as Con(A), but on the logarithmic scale. The use of the logarithmic scale is motivated in the same way as in the case of the Shannon entropy [39]. Function D is clearly a measure of the average conflict among evidential claims within a given body of evidence. Consider, as an example, incomplete information regarding the age of a person X. Assume that the information is expressed by two evidential claims pertaining to the age of X: X is between 15 and 17 years old with degree m(A), where A = {15, 17}, and X is a teenager with degree m(B), where B = {13, 19}. Clearly, the weaker second claim does not conflict with the stronger first claim. Assume that , in this case, the situation is inverted: the claim focusing on B is not implied by the claim focusing on A and, consequently, m(B) dose conflict with m(A) to a degree proportional to number of elements in A that are not covered by B. This conflict is not captured by function Con since in this case.It follows from these observations that the total conflict of evidential claims within a body of evidence {F, m} with respect to a particular claim m(A) should be expressed by function

rather than function Con given by equation (1k). Replacing Con (A) with CON (A), we obtain a new function, which is better justified as a measure of conflict in evidence theory than function D. This new function, which is called strife and denoted by S, is defined by the form,

In ordered possibility distributions , the form of function S (strife) in possibility theory is defined by:

Where U(r) is the measure of possibilistic nonspecificity (U-uncertainty). The maximum value of possibilistic strife, given by equation (1n), depends on n in exactly the same way as the maximum value of possibilistic discord: it increases with n and converges to a constant, estimated as 0.892 as [51]. However the possibility distributions for which the maximum of possibilistic strife are obtained (one for each value of n) are different from those for possibilistic discord. This property and the intuitive justification of function S make this function better candidate for the entropy-like measure in Dempster-shafer theory (DST) than any of the previously considered functions.1.3.4.Applications of Measures of Uncertainty:Uncertainty measures becomes well justified, they can be used in many other contexts as for managing uncertainty and the associated information. For example, they can be used for extrapolating evidence, assessing the strength of relationship between given groups of variables, assessing the influence of given input variables on given output variables, measuring the lost of information when a system is simplified, and the like. In many problem situations, the relevant measures of uncertainty are applicable only in their conditional or relative terms. The use of relevant uncertainty measures is as broad as the use of any relevant measuring instrument [33]. Three basic principles of uncertainty were developed to guide the use of uncertainty measures in different situations. These principles are: a principle of minimum uncertainty, a principle of maximum uncertainty, and a principle of uncertainty invariance.The principle of minimum uncertainty is an arbitration principle. It is used for narrowing down solutions in various systems problems that involve uncertainty. It guides the selection of meaningful alternatives from possible solutions of problems in which some of the initial information is inevitably lost but in different solutions it is lost in varying degrees. The principle states that we should accept only those solutions with the least loss of information i.e. whose uncertainty is minimal. Application of the principle of minimum uncertainty is the area of conflict-resolution problems. For some development of this principle see [38]. For example when we integrate several overlapping models into one larger model, the models may be locally inconsistent. It is reasonable to require that each of the models be appropriately adjusted in such a way that the overall model becomes consistent. It is obvious that some information contained in the given models is inevitably lost by these adjustments. This is not desirable. Hence, we should minimize this lost of information. That is, we should accept only those adjustments for which the total increase of uncertainty is minimal.The principle of maximum uncertainty is essential for any problem that involves ampliative reasoning. Ampliative reasoning is indispensable to science in a variety of ways. For example, when we want to estimate microstates from the knowledge of relevant microstates and partial information regarding the microstates, we must resort to ampliative reasoning. Ampliative reasoning is also common and important in our daily life where the principle of maximum uncertainty is not always adhered to. Its violation leads almost invariable to conflicts in human communication, as well expressed by [93] ..whenever you find yourself getting angry about a difference in opinion, be on your guard; you will probably find, on examination, that your belief is getting beyond what the evidence warrants. This is reasoning in which conclusions are not entailed in the given premises. The principle may be expressed by the following requirement: in any ampliative inference, use all information available, but make sure that no additional information is unwittingly added. The principle of maximum uncertainty is applicable in situations in which we need to go beyond conclusions entailed by verified premises [33]. The principle states that any conclusion we make should maximize the relevant uncertainty within constraints given by the verified premises. In other words, the principle guides us to utilize all the available information but at the same time fully recognize our ignorance. This principle is useful, for example, when we need to reconstruct an overall system from the knowledge of some subsystems. The principle of maximum uncertainty is well developed and broadly utilized within classical information theory, [53] where it is called the principle of maximum entropy [37]. The last principle, the principle of uncertainty invariance, is of relatively recent origin [19]. Its purpose is to guide meaningful transformations between various theories of uncertainty. The principle postulates that the amount of uncertainty should be preserved in each transformation of uncertainty from one mathematical framework to another. The principle was first studied in the context of probability-possibility transformations [50]. Unfortunately, at the time, no well justified measure of uncertainty was available. Due to the unique connection between uncertainty and information, the principle of uncertainty invariance can also be conceived as a principle of information invariance or information preservation [34].1.4.Fuzzy measure: In mathematics,fuzzy measure considers a number of special classes of measures, each of which is characterized by a special property. Some of the measures used in this theory are plausibility and belief measures, fuzzy set membership functionand the classical probability measures. In the fuzzy measure theory, the conditions are precise, but the information about an element alone is insufficient to determine which special classes of measure should be used [44]. The central concept of fuzzy measure theory is the fuzzy measure which was introduced by Choquet in 1953 by [44] and independently defined by Sugeno in 1974 by [72] in the context of fuzzy integrals.Consider, however, the jury members for a criminal trial who are uncertain about the guilt or innocence of the defendant. The uncertainty in this situation seems to be of a different type; the set of people who are guilty of the crime and the set of innocent people are assumed to have very distinct boundaries. The concern, therefore, is not with the degree to which the defendant is guilty, but with the degree to which the evidence proves his membership in either the crisp set of guilty people or the crisp set of innocent people. We assume that perfect evidence would point to full membership in one and only one of these sets. However, our evidence is rarely, if ever, perfect, and some uncertainty usually prevails. In order to represent this type of uncertainty, we could assign a value to each possible crisp set to which the element in question might belong. This value would indicate the degree of evidence or certainty of the elements membership in the set. Such a representation of uncertainty is known as a fuzzy measure.In classical, two-valued logic, we would have to distinguish cold from not cold by fixing a strict changeover point. We might decide that anything below 8 degrees Celsius is cold, and anything else is not cold. This can be rather arbitrary. Fuzzy logic lets us avoid having to choose an arbitrary changeover point essentially by allowing a whole spectrum of degrees of coldness. A set of temperature like hot or cold is represented by a function. Given a temperature, the function will return a number representing the degree of membership of that set. This number is called a fuzzy measures.1.4.1Example:Some fuzzy measure for the set cold:Temperature in 0CFuzzy Measure

-2731 Cold

-401 Cold

00.9 Not quite cold

50.7On the cold side

100.3A bit cold

150.1Barely cold

1000Not cold

10000Not cold

The basis of the idea of coldness may be how people use the word cold perhaps. 30% of people think that 100C is cold (function value 0.3) and 90% think that 00C is cold (function value 0.9). Also it may depend on the context. In terms of the weather, cold means one thing. In terms of the temperature of the coolant in a nuclear reactor cold may means something else, so we would need to have a cold function appropriate to our context.1.4.2Definition:Given a universal set X and a non empty family of subsets of X, a fuzzy measure on X, is a function that satisfies the following requirements by [37]:[1]Boundary requirements: [2]Monotonicity:If .[3]Continuity from below: For any increasing sequence

[4]Continuity from above: For any decreasing sequence

The boundary requirements [1] state that the element in question definitely does not belong to the empty set and definitely does belong to the universal set. The empty set does not contain any element hence it cannot contain the element of our interest, either; the universal set contains all elements under consideration in each particular context; therefore it must contain our element as well.Requirement [2] states that the evidence of the membership of an element in a set must be at least as great as the evidence that the element belongs to any subset of that set. Indeed with some degree of certainty that the element belongs to a set, then our degree of certainty that is belongs to a larger set containing the former set can be greater or equal, but it cannot be smaller. Requirements [3] and [4] are clearly applicable only to an infinite universal set. They can therefore be disregarded when the universal set is finite. Fuzzy measures are usually defined on families that satisfy appropriate properties (rings, semirings, - algebras, etc.). In some cases, consists of the full power set P(X) [37].Three additional remarks regarding this definition are needed. First, functions that satisfy [1], [2], and either [3] or [4] are equally important in fuzzy measure theory as functions that satisfy all four requirements. These functions are called semicontinuous fuzzy measure; they are either continuous from below or continuous from above. Second, it is sometimes needed to generalize fuzzy measures by extending the range of function from [0, 1] to the set of nonnegative real numbers and by excluding the second boundary requirement, . These generalizations are not desirable for our purpose. Third, fuzzy measures are generalizations of probability measures or generalizations of classical measures. The generalization is obtained by replacing the additivity requirement with the weaker requirements of monotonicity and continuity or, at least, semicontinuity [106]. 1.4.3Properties of fuzzy measures: For any , a fuzzy measure is1.Additive:if ;2.super additive:if;3.sub additive: if;4.super modular:if5.sub modular:if ;6.Symmetric:7.Boolean:Let A and B are any two sets then . The monotonicity of fuzzy measures that every fuzzy measures g satisfies the inequality for any three sets ,(1o)Similarly for any two sets, the monotonicity of fuzzy measures implies that every fuzzy measure g satisfies the inequality for any three sets ,(1p)Understanding the properties of fuzzy measures is useful in application. When a fuzzy measure is used to define a function such as the Sugeno integralor Choquet integral, these properties will be crucial in understanding the function's behavior. For instance, the Choquet integral with respect to an additive fuzzy measure reduces to the Lebesgue integral [47]. In discrete cases, a symmetric fuzzy measure will result in the ordered weighted averaging(OWA) operator. Submodular fuzzy measures result in convex functions, while supermodular fuzzy measures result in concave functions when used to define a Choquet integral [72].Since fuzzy measures are defined on the power set(or, more formally, on the -algebraassociated withX), even in discrete cases the number of variables can be quite high. For this reason, in the context of multi-criteria decision analysisand other disciplines, simplification assumptions on the fuzzy measure have been introduced so that it is less computationally expensive to determine and use. For instance, when it is assumed the fuzzy measure isadditive, it will hold that

and the values of the fuzzy measure can be evaluated from the values onX. Similarly, asymmetricfuzzy measure is defined uniquely by values. Two important fuzzy measures that can be used are the Sugeno--fuzzy measure and -additive measures, introduced by Sugenoand Grabisch respectively [66], [68]. Fuzzy measure theory is of interest of its three special branches: probability theory, evidence theory and possibility theory. Although our principle interest is in possibility theory and its comparison with probability theory, evidence theory will allow us to examine and compare the two theories from a broader perspective.1.4.4Probability Theory:Probability represents a unique encoding of incomplete information. The essential task of probability theory is to provide methods for translating incomplete information into this code. The code is unique because it provides a method, which satisfies the following set of properties for any system (95).1. If a problem can be solved in more than one way, all ways must lead to the same answer.2. The question posed and the way the answer is found must be totally transparent. There must be no laps of faith required to understand how the answer followed from the given informations.3. The methods of solution must not be ad-hoc. They must be general and admit to being used for any problem, not just a limited class of problems. Moreover the applications must be totally honest. For example, it is not proper for someone to present incomplete information, develop an answer and then prove that the answer is incorrect because in the light of additional information a different answer is obtained.4. The process should not introduce information that is not present in the original statement of the problem.1.4.4.1Basic Terminologies Used in Probability Theory:(I)The Axioms of Probability: The language systems, contains a set of axioms that are used to constrain the probabilities assigned to events [95]. Four axioms of probability are as follows:1. All values of probabilities are between zero and one i.e.

2. Probabilities of an event that are necessarily true have a value of one, and those that are necessarily false have a value of zero i.e. P(True) = 1 and P(False) = 0.3. The probability of a disjunction is given by:4. A probability measure, Pro is required to satisfy the equation .This requirement is usually referred to as the additivity axiom of probability measures [36].An important result of these axioms is calculating the negation of a probability of an event. i.e..(II)The Joint Probability Distribution: The joint probability distribution is a function that specifies a probability of certain state of the domain given the states of each variable in the domain. Suppose that our domain consists of three random variables or atomic events [95]. Then an example of the joint probability distribution of this domain where would be same value depending upon the values of different probabilities of. The joint probability distribution will be one of the many distributions that can be calculated using a probabilistic reasoning system [50].(III)Conditional Probability and Bayes Rule:Suppose a rational agent begins to perceive data from its world, it stores this data as evidence. This evidence is used to calculate a posterior or conditional probability which will be more accurate than the probability of an atomic event without this evidence, known as a prior or unconditional probability [95].We take example to define bayes rule; the reliability of a particular skin test for tuberculosis (TB) is as follows:If the subject has TB then the sensitivity of the test is 0.98.If the subject does not have TB then the specificity of the test is 0.99.From a large population, in which 2 in every 10,000 people have TB, a person is selected at random and given the test, which comes back positive. What is the probability that the person actually has TB?Lets define event A as the person has TB and event B as the person tests positive for TB. It is clear that the prior probability.The conditional probability,the probability that the person will test positive for TB given that the person has TB. This was given as 0.98. The other value we need , the probability that the person will test positive for TB given that the person does not have TB. Since a person who does not have TB will test negative 99% (given) of the time, he/she will test positive 1% of the time and therefore . By Bayes Rule as:

We might find this hard to believe, that fewer than 2% of people who test positive for TB using this test actually have the disease. Ever though the sensitivity and specificity of this test are both high, the extremely low incidence of TB in the population has a tremendous effect on the tests positive predictive value, the population of people who test positive that actually have the disease. To see this, we might try answering the same question assuming that the incidence of TB in the population is 2 in 100 instead of 2 in 10,000.(IV)Conditional Independence:When the result of one atomic event, A, does not affect the result of another atomic event, B, those two atomic events are known to be independent of each other and this helps to resolve the uncertainty [13]. This relationship has the following mathematical property: . Another mathematical implication is that: Independence can be extended to explain irrelevant data in conditional relationship.1.4.4.2Disadvantages with Probabilistic Method:Probabilities must be assigned even if no information is available and assigns an equal amount of probability to all such items. Probabilities require the consideration of all available evidence, not only from the rules currently under consideration [50]. Probabilistic methods always require prior probabilities which are very hard to found out apriority [36]. Probability may be inappropriate where as the future is not always similar to the past. In probabilistic method independence of evidences assumption often not valid and complex statements with conditional dependencies cannot be decomposed into independent parts. In this method relationship between hypothesis and evidence is reduced to a number [13]. Probability theory is an ideal tool for formalizing uncertainty in situations where class frequencies are known or where evidence is based on outcomes of a sufficiently long series of independent random experiments [95].1.4.5Evidence Theory:Dempster-Shafer theory (DST) is a mathematical theory of evidence. The seminal work on the subject is done by Shafer [43], which is an expansion of previous work done by Dempster [4]. In a finite discrete space, DST can be interpreted as a generalization of probability theory where probabilities are assigned to sets as opposed to mutually exclusive singletons. In traditional probability theory, evidence is associated with only one possible event. In DST, evidence can be associated with multiple possible events, i.e. sets of events. DST allows the direct representation of uncertainty. There are three important functions in DST by [4], [43]: (1) The basic probability assignment function (bpa or m), (2) The Belief function (Bel) and (3) The Plausibility function (Pl). The theory of evidence is based on two dual non-additive measures: (2) and (3).(1) Basic Probability Assignment:Basic probability assignment does not refer to probability in the classical sense. The basic probability assignment, represented by m, defines a mapping of the power set to the interval between 0 and 1, s.t. satisfying the following properties:[50]. (a) , and

The value of m(A) pertains only to the set A and makes no additional claim about any subset of A. Any further evidence on the subset of A would be represented by another basic probability assignment. The summation of the basic probability assignment of all the subsets of the power set is 1. As such, the basic probability assignment cannot be equated with a classical probability in general [13].(2) Belief Measure: Given a measurable space, a belief measure is a function satisfying the following properties:(a) ,(b) , and

Due to the inequality (1r), belief measures are called superadditive. When X is infinite, function Bel is also required to be continuous from above. For each is defined as the degree of belief, which is based on available evidence [4], that a given element of X belongs to the set Y. The inequality (1r) implies the monotonicity requirement [2] of fuzzy measure. Let

Applying now to (1r), we get

Since , we have

Let in (1r) for n=2. Then we have,

(1s)Inequality (1s) is called the fundamental property of belief measures.(3) Plausibility Measure: Given a measurable space, a plausibility measure is a function satisfying the following properties:(a) ,(b) , and

Due to the inequality (1s), plausibility measures are called subadditive. When X is infinite, function Pl is also required to be continuous from below [43].Let in (1t) for n=2. Then we have,

(1u)According to inequality (1s) and (1u) we say that each belief measure, Bel, is a plausibility measure, Pl, i.e. the relation between belief measure and plausibility measure is defined by the equation [59].(1v)(1w)1.4.6Possibility Theory:Possibility theory is an uncertainty theory devoted to the handling of incomplete information. It is comparable to probability theory because it is based on set-functions. Possibility theory has enabled a typology of fuzzy rules to be laid bare, distinguishing rules whose purpose is to propagate uncertainty through reasoning steps, from rules whose main purpose is similarity-based interpolation [15]. The name Theory of Possibility was coined by Zadeh [59], who was inspired by a paper by Gaines and Kohout [8]. In Zadeh's view, possibility distributions were meant to provide a graded semantics to natural language statements. Possibility theory was introduced to allow a reasoning to be carried out on imprecise or vague knowledge, making it possible to deal with uncertainties on this knowledge. Possibility is normally associated with some fuzziness, either in the background knowledge on which possibility is based, or in the set for which possibility is asserted [36].Let S be a set of states of affairs (or descriptions thereof), or states for short. A possibility distribution is a mapping from S to a totally ordered scale L, with top 1 and bottom 0, such as the unit interval. The function represents the state of knowledge of an agent (about the actual state of affairs) distinguishing what is plausible from what is less plausible, what is the normal course of things from what is not, what is surprising from what is expected [70]. It represents a flexible restriction on what is the actual state with the following conventions (similar to probability, but opposite of Shackle's potential surprise scale):(1) (s) = 0 means that state s is rejected as impossible;(2) (s) = 1 means that state s is totally possible (= plausible).For example, imprecise information such as Xs height is above 170cm implies that any height h above 170 is possible any height equal to or below 170 is impossible for him. This can be represented by a possibility measure defined on the height domain whose value is 0 if and 1 if (0 = impossible and 1 = possible).when the predicate is vague like in X is tall, the possibility can be accommodate degrees, the largest the degree, the largest the possibility. For consonant body of evidence, the belief measure becomes necessity measure and plausibility measure becomes possibility measure.Hence Becomes And Becomes And also Some other important measures on fuzzy sets and relations are defined here.1.4.7Additive Measure:Let X, be a measurable space. A function is an - additive measure when the following properties are satisfied: 1. 2. If n = 1, 2, ... is a set of disjoint subsets of then

The second property is called -additivity, and the additive property of a measurable space requires the -additivity in a finite set of subsets . A well-known example of -additive is the probabilistic space (X, , p) where the probability p is an additive measure such that for all subsets . Other known examples of -additive measure are the Lebesgue measures defined that are an important base of the XX century mathematics [63]. The Lebesgue measures generalise the concept of length of a segment, and verify that if

Other measures given by Lebesgue are the exterior Lebesgue measures and interior Lebesgue measures. A set A is Lebesgue measurable when both interior and exterior Lebesgue measures are the same [67]. Some examples of Lebesgue measurable sets are the compact sets, the empty set and the real numbers set R.1.4.8 - additive fuzzy measureA discrete fuzzy measure on a setXis called -additive if its Mbius representation verifies , whenever for any , and there exists a subset F with elements such that.The -additive fuzzy measure limits the interaction between the subsets to size. This drastically reduces the number of variables needed to define the fuzzy measure, and as can be anything from 1 to, it allows for a compromise between modelling ability and simplicity [66].1.4.9Normal MeasureLet X, be a measurable space. A measure is a normal measure if there exists a minimal set and a maximal set in such that:1. 2. For example, the measures of probability on a space X, are normal measures with and . The Lebesgue measures are not necessarily norma [63]. 1.4.10Sugeno Fuzzy Measure:Let be an -algebra on a universe X. A Sugeno fuzzy measure is verifying by [99]:1. 2.If .3.If and then

Property 3 is called Sugenos convergence. The Sugeno measures are monotone but its main characteristic is that additivity is not needed. Several measures on finite algebras, as probability, credibility measures and plausibility measures are Sugeno measures. The possibility measure on possibility distributions introduced by Zadeh [59] gives Sugeno measures. 1.4.11Fuzzy Sugeno -MeasureSugeno [72] introduces the concept of fuzzy -measure as a normal measure that is -additive. So the fuzzy -measures are fuzzy measures.Let and let X, be a measurable space. A function is a fuzzy -measure if for all disjoint subsets A, B in , . For example, if then the fuzzy -measure is an additive measure.1.5.Null-additive fuzzy measure:The concept of fuzzy measure does not require additivity , but it requires monotonicity related to the inclusion of sets [45]. Additivity can be very effective and convenient in some applications, but can also be somewhat inadequate in many reasoning environments of the real world as in approximate reasoning, fuzzy logic, artificial intelligence, game theory, decision making, psychology, economy, data mining, etc., that require the definition of non-additive measures and large amount of open problems [26]. For example, the efficiency of a set of workers is being measured, the efficiency of the some people doing teamwork is not the addition of the efficiency of each individual working on their own. The theory of non-additive set functions had influences on many parts of mathematics and different areas of science and technology. Recently, many authors have investigated different type of non-additive set functions, as sub-measures, k-triangular set functions, - decomposable measures, null-additive set functions and others. The range of the null-additive set functions, introduced by Wang [109]. Suzuki [45] introduced and investigated atoms of fuzzy measures and Pap [26] introduced and discussed atoms of null-additive set functions.1.5.1Definition:Let X be a set and a algebra over X. A set function m, is called null-additive, if: [109].It is obvious that for null-additive set function m we have whenever there exists such that 1.5.2Examples:1. - decomposable measure with respect to a t-conorm , such that and whenever and is always null-additive [25].2.A generalization of -decomposable measure from example 1 is the -decomposable measure . Namely, let [a, b] be a closed real interval, where . The operation (pseudo-addition) [25] is a function which is commutative, non-decreasing, associative and has a zero element, denoted by 0 which is either a or b. A set function is a -decomposable measure if there hold and whenever and . Then m is null-additive.3.Let whenever . Then m is null-additive.4.Let and define m in the following way:Then m is not null-additive.1.5.3Saks decomposition:Let m be a null-additive set function. A set with is called an atom provided that if then [25]1.,or2.1.5.4Remarks:1.For null-additive set functions the condition 2 implies , what is analogous to the property of the usual -finite measures (for this case the opposite implication is also true).2.For -finite measure m, each of its atom has always finite measure.1.5.5Darboux property:A set function m is called autocontinuous from above (below) if for every and every there exists such that [110](resp. whenever holds.1.5.6Remark:If m is a fuzzy measure, then we can omit in the preceding definition the supposition (resp. ).The set function m is a null-additive non fuzzy measure which is autocontinuous from above and from below.The autocontinuity from above (below) in the sense of Wang [110], if

whenever , is called W-autocontinuity. If m is a finite fuzzy measure, then m is autocontinuous from above (below) iff m is W-autocontinuous from above (below). If m is a finite fuzzy measure, then m is autocontinuous from below iff it is autocontinuous from below.Each set function which is autocontinuous from above is null-additive but there exists null-additive fuzzy measure which is not autocontinuous from above (Example 1). If the set X is countable, then null-additivity of a finite fuzzy measure is equivalent to the autocontinuity [109].

Possibility Theory versus Probability Theory in Fuzzy Measure Theory2.1Concept of associative probabilities to a fuzzy measure:Fuzzy Measures are appropriate tools to represent information or opinion state. A method for associating a set of probabilities to each fuzzy measure has been developed [21]. The purpose of concept of associative probabilities to a fuzzy measure is to use this concept to define differences between fuzzy measures and briefly to show how differences could be used for studying some characteristics of fuzzy measures. Let us consider a finite set . A fuzzy measure on X defined in chapter 1 section 1.4. Two fuzzy measures g and g* are called dual fuzzy measures if and only if the following relation is satisfied:. Where is the complement of A.Duality is an important concept, since it permits us to obtain alternative representations of a piece of information: we will consider the situation that two dual fuzzy measures contain the same information, but codified in a different way. The property of the Choquet integral [30] called monotone expectation: the integral of the function h with respect to a fuzzy measure g coincides with the mathematical expectation of h with respect to a probability measure which depends only on g and the ordering of the values of h, was the starting point [21] to associate a set of n! probabilities to each fuzzy measure, in the following way.2.1.1Definition: Let g be a fuzzy measure on X. The probability functions defined by:

(2a)For each are called the associated probabilities to the fuzzy measure g, where is the group of permutation of a set with n elements.From the basic probability assignment the upper and lower bounds of an interval can be defined. This internal contains the precise probability of a set of interest and is bounded by two non-additive continuous measures called Belief and plausibility. The lower bound is Belief and the upper bound is Plausibility. It is possible to obtain the basic probability assignment from the Belief measure with the following inverse function:

where |R-S| is the difference of the cardinality of the two sets.The very general result of the approximation problem has been established by [103] and is not well known in fuzzy literature. Volker [58] will generalize it, following a different line of reasoning which emphasizes the contribution of the inner extension procedures within abstract measure theory. Further application leads to plausibility and possibility measures as upper envelopes of probability measure, even in the context of infinite universes of discourse. Especially this supports the semantically point of view to regard possibility degrees as upper bounds of probabilities an argument proposed by [14], [16].2.1.2Theorem: A Belief measure (Bel) on a finite power set is a probability measure if and only if the associated basic probability assignment function is given by: and for all subsets of X that are not singletons.Proof: Assume that Bel is a probability measure. For the empty set , the theorem trivially holds, since by definition of . Let and assume . Then by repeated application of the additivity axiom:

For all sets such that . We obtain,

Since for any , we have

Hence, Bel is defined in terms of a basic probability assignment that focuses only on singletons. Assume now that a basic probability assignment function is given such that,

Then for any set such that , we have

Consequently, Belief measure is a probability measure. Hence provedAccording to above theorem, probability measures on finite sets are thus fully represented by a function such that . This function is usually called a probability distribution function. Let be referred to as a probability distribution on X. When the basic probability assignment function focuses only on singletons, as required for probability measures, then the Belief measure and Plausibility measure become equal. Hence,

2.1.3Basic Mathematical Properties of Possibility Theory and Probability Theory:1. Probability Theory: It is based on measures of one type: Probability measure (P).Possibility Theory: It is based on measures of two types: (a) Possibility measures , (b) Necessity measures .2. Probability Theory: Here body of evidence consists of singletons.Possibility Theory: Body of evidence consist of a family of nested subsets.3. Probability Theory: Probability measures holds additivity i.e., .Possibility Theory: Possibility measures and Necessity measures follow the max\min rules:.4. Probability Theory: Unique representation of P by a Probability distribution function via the formula

Possibility Theory: Unique representation of by a Possibility distribution function via the formula

5. Probability Theory: It is normalized by

Possibility Theory: It is normalized by

6. Probability Theory: Total ignorance:for all Possibility Theory: Total ignorance:

7. Probability Theory: Possibility Theory:

As obvious from their mathematical properties, possibility, necessity and probability measures do not overlap with one another except for one very special measure, which is characterized by only one focal element, a singleton [37]. The two distribution functions that represent probabilities and possibilities become equal for this measure: one element of the universal set is assigned the value of 1, with all other element being assigned a value of 0. This is clearly the only measure that represents perfect evidence.2.1.4Difference between probability theory and possibility theory: [22]1. The theory of possibility is analogous to, yet conceptually different from the theory of probability. Probability is fundamentally a measure of the frequency of occurrence of an event, while possibility is used to quantify the meaning of an event.2. Value of each probability distribution are required to add to 1, while for possibility distributions the largest values are required to be 1.3. Probability theory is an ideal tool for formalizing uncertainty in situations where class frequencies are known or where evidence is based on outcomes of a sufficiently long series of independent random experiments. On the other hand possibility theory is ideal for formalizing incomplete information expressed in terms of fuzzy propositions.4. Possibility measures replace the additivity axiom of probability with the weaker subadditivity condition.5. Probabilistic bodies of evidence consist of singletons, while possibilistic bodies of evidence are families of nested set.6. A difference between the two theories is in their expressions of total ignorance. In probability theory, total uncertainty is expressed by the uniform probability distribution on the universal set: for all . In possibility theory, it is expressed in the same way as in evidence theory .7. Possibility measures degree of ease for a variable to be taken a value where as probability measures the likelihood for a variable to take a value.8. Possibility theory are still less developed than their probabilistic counterparts, it is already well established that possibility theory provides a link between fuzzy sets and probability theory are connected with probability theory.2.1.5Similarity between probability theory and possibility theory: [22]1. When information regarding some phenomenon is given in both probabilistic and possibilistic terms, the two descriptions should be in some sense consistent. That is, given a probability measure (P) and a possibility measure both define on , the two measures should satisfy some consistency condition.2. Possibility theory and probability theory are suitable for modelling certain type of uncertainty and suitable for modelling other types.3. Notion of non-interactiveness on possibility theory is analogous to the notion of independence in probability theory. If two random variables x and y are independent, their joint probability distribution is the product of their individual distributions. Similarly if two linguistic variables are non-interactive, their joint probabilities are formed by combining their individual possibility distribution through a fuzzy conjunction operator.4. Possibility theory may be interpreted in terms of interval-valued probabilities, provided that the normalization requirement is applied. Due to the nested structure of evidence, the intervals of estimated probabilities are not totally arbitrary. If , then the estimated probabilities are in the interval ; if , then the estimated probabilities are in the interval . Due to these properties, belief measures and plausibility measures may be interpreted as lower and upper probability estimates.There are multiple interpretations of probability theory and possibility theory. Viewing necessity and possibility measures as lower and upper probabilities opens a bridge between the two theories, which allow us to adjust some of the interpretations of probability theory to the interval-valued probabilities of possibilistic type.There are two basic approaches to possibility/probability transformations, which both respect a form of probability-possibility consistency. One, due to [32], [50] is based on a principle of information invariance, the other [17] is based on optimizing information content. Klir assumes that possibilistic and probabilistic information measures are commensurate. The choice between possibility and probability is then a mere matter of translation between languages neither of which is weaker or stronger than the other [36]. It suggest that entropy and imprecision capture the same facet of uncertainty, albeit in different guises.2.2Shannon entropy of fuzzy measure:Discuss some variants of fuzzy subsets entropies and the linking relations of these quantities with indicate a similarity of many important characteristics of fuzzy subsets entropies and Shannons entropy.A function which forms the basis of classical information theory measures the average uncertainty associated with the prediction of outcomes in a random experiment; its range is . It is denoted by H and defined as:

and it is satisfies the following conditions:1.,when for some .2.,when for all .When defines the uniform probability distribution on X then the function is called Shannon entropy, which is applicable only to probability measures in evidence theory. The function H was proposed by Shannon [10]. It was proven in numerous ways, from several well-justified axiomatic characterisations, that this function is the only sensible measure of uncertainty in probability theory. The Shannon entropy is a measure of the average information content one is missing when one does not know the value of the random variable. It represents an absolute limit on the best possible lossless compression of any communication under certain constraints, treating messages to be encoded as a sequence of independent and identically distributed random variables.Since value are required to add to 1 for all , (2.6) can be rewritten as

The choice of the logarithmic function is a result of the axiomatic requirement that the joint uncertainty of several independent random variables be equal to the sum of their individual uncertainty. From equation (2f) the Shannon entropy is the mean value of the conflict among evidential claims within a given probabilistic body of evidence. The information that we receive from an observation is equal to the degree to which uncertainty is reduced.Let us perform a splitting of the set X point by point. In this case Shannons entropy of probability distribution , turns into where,,, is the membership function of a fuzzy point and is its dual, i.e. , . If we avail the branching property of function H [2], then:

And (2j) is Zadehs entropy [61], i.e. an entropy of fuzzy set X with respect to probability distribution . And also where,

Function is actually a kullback directed divergence [94] and

is a weighted entropy of De Luca and Termini [1].Equality (2g) is Hirotos measure of uncertainty [3]. Notice that if we avail the branching property in some other way then we rewrite (2g) in following form:

In addition functions and are connected by Jumaries entropy [41].(2p)Formulas (2h) and (2j) represent particular cases of Zadehs and De Luca and Terminis entropies, respectively. The characteristic equations of and are particular cases of a more general equation whose solutions areentropies; they are connected, with the usual entropy of probability distribution as (2j) and (2m) with Shannons entropy of split distribution [38].

is defined as

is a of Zadeh and

is the weighted of De Luca and Termini. Consequently,

Definitions analogous to (2q) (2t) can be given for dual subsets.Properties of Zadehs entropy and weighted entropy which will be considered below are represented as inequalities. These are consequence of the relation between the arithmetical and geometrical means which in logarithmic representation has the form [2]:

And which is also a basic inequality when one considers the Shannons entropy.2.2.1 Theorem: Let be a fuzzy subset with values of membership function and a probability distribution on X, then:

Proof: Equation (2v) is a direct consequence of inequality (2u). Assuming:

Where equality holds if and only if i.e. all are equal, . Hence proved.Recently, Marichal [52] the notion of entropy of a discrete Choquet capacity as a generalization of the Shannon entropy and showed that it satisfies many properties that one would intuitively require from such a measure.Let us consider N be a finite set of n natural numbers i.e. and be a power set of N. Again let be a discrete Choquet capacity or discrete fuzzy measure on N. The Choquet capacity is said to be normalized if hence denote the set of all normalized Choquet capacity on N. Let S be any subset of N as can be interpreted as the degree of importance of S, then generalized entropy is defined as:

Where Hence equation (2x) is the generalized form of Shannon entropy.For particular fuzzy measures, such as belief and plausibility measures, two entropy-like measures were proposed in evidence theory in the early 1980s, namely the measure of dissonance and the measure of confusions [16]. For general fuzzy measures it seems that no definition of entropy was given until 1999 when two proposals were introduced successively by Marichal [52] and Yager [88]. These definitions are very similar but have been introduced independently and within completely different frameworks. The first one gives the degree to which numerical values are really used when aggregated by a Choquet integral. The second one measures the amount of information in a fuzzy measure when it is being used to represent the knowledge about an uncertain variable.Suppose that represents a set of criteria in a multi-criteria decision making problem and consider a fuzzy measure on N and suppose that represent quantitative evaluations of an object with respect to criteria respectively. We assume that these evaluations are defined on the same measurement scale. A global evaluation (average value) of this object can be calculated by means of the Choquet integral with respect to . Formally the Choquet integral of with respect to a fuzzy measure on N is defined by,

such that , also and . It would be interesting to appraise the extent to which the argumentsare really used in the aggregation by the Choquet integral. This extent, which depends only on the importance of criteria, can be measured by the following entropy-like function, proposed and justified by Marichal [52]. We call it lower entropy. The lower entropy of a fuzzy measure on N is defined by,

Consider a variable V whose the exact value, which lies in the space , is not completely known. In many situations, the best we can do is to formulate our knowledge about V by means of a fuzzy measure on N. For each subset of values, represents a measure associated with our belief (or the confidence we have) that the value of V is contained in the subset S. Here monotonicity of means that we cannot be less confident that V lies in a smaller set than a larger one.Now, consider the Shapley value [62] of , i.e., the vector,

whose i-th component, called Shapley index of , is defined by:

It can be easily proved that the indices always sum up to one, so that the Shapley value of any fuzzy measure on N is a probability measure on N. From this observation, Yager [88] proposed to evaluate the uncertainty associated with the variable V by taking the Shannon entropy of the Shapley value of . This leads to the upper entropy. The upper entropy of a fuzzy measure on N is defined by,, that is,

2.3Basic properties of entropy of fuzzy measure:Here we analyze some basic properties of entropy of fuzzy measure. The main properties of entropy of fuzzy measure, we have:2.3.1Continuity:The measure H should be continuous, in the sense that changing the values of the probabilities by a very small amount, should only change the H value by a small amount.2.3.2Maximality:The maximality property for the Shannon entropy says that the uncertainty of the outcome of an experiment is maximum when all outcomes have equal probabilities, i.e. the uncertainty is highest when all the possible events are equiprobable. Thus for any probability measure on N,.And the entropy will increase with the number of outcomes,.More precisely, reaches its maximal value (log n) if and only if is the uniform distribution on N. concerning the entropies , we have the following result,. Moreover:(i) if and only if for all .(ii) if and only if .The second inequality and property (ii) follow from above property of the Shannon entropy.2.3.3Additivity:The amount of entropy should be independent of how the process is considered, as being divided into parts. Such a functional relationship characterizes the entropy of a system with respect to the sub-systems. It demands that the entropy of every system can be identified and then computed from the entropies of their sub-systems. i.e. if

This is because statistical entropy is a probabilistic measure of uncertainty or ignorance about data, whereas information is a measure of a reduction in that uncertainty.2.3.4Symmetry:It is obvious that the Shannon entropy H is symmetric in the sense that permuting the elements of N has no effect on the entropy. This symmetry property is actually also fulfilled by and . For any permutation on , we denote by the fuzzy measure on N defined by for all , where . We then have the following result. .This accord with the interpretation of the entropy in the aggregation framework. Indeed, one can easily show that.Thus, permuting the arguments of the Choquet integral has no effect on the degree to which one uses these arguments. We also have the uncertainty associated with the variable V is independent of any permutation of elements of N.2.3.5Expansibility: The classical expansibility property for the Shannon entropy says that suppressing an outcome with zero probability does not change the uncertainty of the outcome of an experiment [54]:.This property can be extended to the framework of fuzzy measures in the following way. Let be a fuzzy measure on N and let be a null element for , that is such that for all . Denote also by the restriction of to. We then have the following result, if is a null element for then,.This is a very natural property in the aggregation framework. Sinceelement does not contribute in the decision problem, it can be omitted without changing the result [99],.Whenever is a null element for .

2.3.6Decisivity:For any probability measure on N, we clearly have . Now, the decisivity property for the Shannon entropy says that there is no uncertainty in an experiment in which one outcome has probability one [54]:.More precisely, reaches its minimal value (0) if and only if is a Dirac measure on N. Concerning the entropies and , we observe two different behaviours, as the following result shows, . Moreover,(i) if and only if is a binary-valued fuzzy measure.(ii) if and only if is a Dirac measure.Property(i) is quite relevant for the aggregation framework. Indeed, it can be shown [52] that is a binary-valued fuzzy measure if and only if .In other terms, is minimum (0) if and only if only one argument is really used in the aggregation, this is a fundamental condition.

2.3.7Increasing Monotonicity:Let and define by

Then for any , we have

We now state another very important property of which follows by the equation,

is called strict concavity of Shannon entropy.2.3.8Strict Concavity:For any and , we have

An immediate consequence of the previous property is that maximizing over a convex subset of always leads to a unique global maximum.For probability distributions, the strict concavity of the Shannon entropy and it naturalness as a measure of uncertainty gave rise to the maximum entropy principle, which was stated by [52] as follows:When one has only partial information about the possible outcomes of a random variable, one should choose its probability distribution so as to maximize the uncertainty about the missing information.In other words, all the available information should be used, but one should be a uncommitted as possible about missing information. In more mathematical terms, this principle states that among all the probability distributions that are in accordance with the available prior knowledge (that is as set of constraints), one should choose the one that has maximum uncertainty.2.4.Applications of entropy of fuzzy measure:The study of different concepts of entropy will be very interesting, and not only on physics, but also on information theory, and other mathematical sciences as fuzzy measures, considered in its more general vision. Also it may be a very useful tool for bio-computing, or in many others, such as studying environmental sciences. This is because, among other interpretations with important practical consequences, the law of entropy means that energy cannot be fully recycled. Many quotations have been made until now referring to the content and significance of this fuzzy measure, for example: Gain in entropy always means loss of information, and nothing more. [40]. Information is just known entropy. Entropy is just unknown information. [71]. Mutual information and relative entropy, also called Kullback-Leibler divergence, among other related concepts, have been very useful in learning systems, both in supervised and unsupervised cases.Entropy and related information measures provide descriptions of the long term behaviour of random processes [5], and that this behaviour is a key factor in developing the Coding Theorems of IT (Information Theory). The contributions of Andrei Nikolaievich Kolmogorov to this mathematical theory provide great advances to the Shannon formulations, proposing a new complexity theory, now translated to Computer Sciences. According to such theory, the complexity of a message is given by the size of the program necessary to enable the reception of such a message. From these ideas, Kolmogorov analyzes the entropy of literary texts and the subject Pushkin poetry. Such entropy appears as a function of the semantic capacity of the texts, depending on factors such as their extension and also the flexibility of the corresponding language. It should also be mentioned that Norbert Wiener, considered the founder of Cybernetics, who in 1948 proposed a similar vision for such a problem. However, the approach used by Shannon differs from that of Wiener in the nature of the transmitted signal and in the type of decision made by the receiver. In the Shannon model, messages are firstly encoded, and then transmitted, whereas in the Wiener model the signal is communicated directly through the channel without need of being encoded. Another measure conceptualized by R. A. Fischer, the so called Fisher Information (FI), applies statistics to estimation, representing the amount of information that a message carries concerning an unobservable parameter.Certainly the initial studies on IT were undertaken by Harry Nyquist in, and later by Ralph Hartley, who recognized the logarithmic nature of the measure of information. This was later essential the key in Shannon and Wieners papers. The contribution of the Romanian mathematician and economist Nicholas Georgescu-Roegen, is also very interesting, whose great work was The Entropy Law and the Economical Process. In this memorable book, he proposed that the second law of thermodynamics also governs economic processes. Such ideas permitted the development of some new fields, such as Bioeconomics or Ecological Economics. Also some others should be noted, studying a different kind of measure, the so called inaccuracy measure, involving two probability distributions. R. Yager [89], and M. Higashi and G. J. Klir [69] showed the entropy measure as the difference between two fuzzy sets. More specifically, this is the difference between a fuzzy set and its complement, which is also a fuzzy set.

Properties of Null-Additive and Absolute Continuity of a Fuzzy Measure3.1Absolute continuity of a fuzzy measure: In Calculus, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. In classical measure theory [79], if is a measure space and f is a nonnegative integrable function, then the Lebesgue integral,

Defines a measure N on that is absolutely continuous with respect to M in the sense that,

The absolute continuity of N with respect to M is usually denoted by . When N is finite, then absolute continuity can also be defined as,

or alternatively, as for any there exists such that whenever and The notion of absolute continuity allows one to obtain generalisations of the relationship between the two central operations of calculus, differentiation and integration, expressed by the fundamental theorem of calculus in the framework of Riemann integration. Such generalisations are often formulated in terms of Lebesgue integration.Similarly, if is a fuzzy measure space [106] and f is a nonnegative measurable function then the fuzzy integral [46], [72],

defines a lower semi-continuous fuzzy measure on and moreover if is finite, then is a finite fuzzy measures [106]In general let denote a measurable space and let M and N denote fuzzy measures (or semicontinuous fuzzy measures) on . The absolute continuity of N with respect to M is denoted by of type and the absolute continuity of classical measures can be generalized for fuzzy measures in different ways which as follows: 3.1.1Definition:(1) iff : , .(2) iff .(3) iff is non-increasing, .(4) iff .(5) iff .(6) iff or iff for any there exists such that (7) iff (8) iff (9) iff or iff for any there exists such that Where and the colon(:) symbol stands for whenever (when appropriate, we include equivalent formulations). Type (1) and (6) in definition 1 has been used in classical measure theory as the definition of absolute continuity, where they coincide when N is finite. Type (9) has been employed for an extension theorem of fuzzy measures [107]. All types of absolute continuity given in definition 3.1.1 coincide with the classical definition of absolute continuity when both M and N are classical measures (i.e. additive) and N is finite. Now we define varieties of type 2, 4, 5, 7, 8 and 9 of absolute continuity, which are denoted by the subscripts a and b. Otherwise the same notation is used as in definition 3.1.1.3.1.2Definition:(1) or iff or iff or iff .(2) or iff , or iff .(3) iff or iff .(4) iff or iff .(5) iff or iff .(6) iff , or iff .(7) iff or iff .(8) iff or iff .(9) iff or iff .(10) iff or iff .(11) iff for anythere exists such that or iff for anythere exists such that .(12) iff for anythere exists such that .In general these relations associated with the varieties of absolutely continuity introduced in definition 3.1.2 are neither reflexive nor transitive. All types of absolute continuity given in this section can be regarded as generalizations of the classical concept of absolute continuity. A theorem for absolute continuity of a fuzzy measure is defined as:3.1.3Theorem: Let M and N be two finite fuzzy measures such that they are continuous from above and continuous from below. If N is auto-continuous from above, then M is absolutely continuous with respect to N iff M is absolutely continuous with respect to N.Proof: It is obvious that if M is absolutely continuous with respect to N, then M is absolutely continuous with respect to N.Suppose now that . If the theorem would not be true, then there would exist and a sequence from such that where n={1,2,3,.........}(3e)Since N is auto- continuous from above there exist a sub-sequence of the sequence such that

By the continuity from above of N we have,

Since N is continuous from below we obtain by equation (3.6),

Hence by equation (3g),

On the other hands, we obtain by the continuity from above and continuity from below of the fuzzy measure M and equation (3e)

Which contradict our assumption, hence proved this theorem.3.2Some results on null additive fuzzy measure:A set function , is called null additive, if we have whenever [28]. Some results on null additive fuzzy measures are as follows. A set function is said to be(1) Weakly null additive, if whenever .(2) Converse null additive, if whenever .(3) Pseudo null additive, if whenever .(4) If is null additive, then it is weakly null additive. If is pseudo null additive, then it is converse null additive.(5) A non additive measure is said to be - null additive, if for every sequence of pairwise disjoint sets from and

(6) is - null additive if and only if is null additive and implies

for every sequence of pairwise disjoint sets from .For the case of non additive measure theory, the situation is not so simple. There are many discussions on Lebesgue decomposition type theorem such as, a version on submeasure [82], a version on - decomposition measure [25], a version on finite fuzzy measure [81] and a version on signed fuzzy measure [46] and so on.A non additive measure on is an real valued set function is said to be,

is singular with respect to , and denote . Let and be two finite fuzzy measures. If , then is absolutely continuous with respect to [107]. On the other hand, if for every there is a such that then we say that is absolutely continuous with respect to . Now we define a theorem,3.2.1Theorem: Let be a null additive fuzzy measure which is continuous from above and continuous from below. Then there exist a set A from such that, , .Proof: Let a sequence from which will generate the desired set A. Let , we take A1 from such that,

This is possible by the continuity from below of . We choose A2 from such that,

Repeating this procedure, we choose a sequence such that,

holds. We take

Then by the construction equation (3h) holds. The continuity from above of implies