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An introduction to measure theoryTerence TaoDepartment of Mathematics, UCLA, Los Angeles, CA90095E-mail address: tao@math.ucla.eduTo Garth Gaudry, who set me on the road;To my family, for their constant support;And to the readers of my blog, for their feedback and contributions.ContentsPreface ixNotation xAcknowledgments xviChapter 1. Measure theory 11.1. Prologue: The problem of measure 21.2. Lebesgue measure 171.3. The Lebesgue integral 461.4. Abstract measure spaces 791.5. Modes of convergence 1141.6. Dierentiation theorems 1311.7. Outer measures, pre-measures, and product measures 179Chapter 2. Related articles 2092.1. Problem solving strategies 2102.2. The Radamacher dierentiation theorem 2262.3. Probability spaces 2322.4. Innite product spaces and the Kolmogorov extensiontheorem 235Bibliography 243viiviii ContentsIndex 245PrefaceIn the fall of 2010, I taught an introductory one-quarter course ongraduate real analysis, focusing in particular on the basics of mea-sure and integration theory, both in Euclidean spaces and in abstractmeasure spaces. This text is based on my lecture notes of that course,which are also available online on my blog terrytao.wordpress.com,together with some supplementary material, such as a section on prob-lem solving strategies in real analysis (Section 2.1) which evolved fromdiscussions with my students.This text is intended to form a prequel to my graduate text[Ta2010] (henceforth referred to as An epsilon of room, Vol. I ),which is an introduction to the analysis of Hilbert and Banach spaces(such as Lpand Sobolev spaces), point-set topology, and related top-ics such as Fourier analysis and the theory of distributions; together,they serve as a text for a complete rst-year graduate course in realanalysis.The approach to measure theory here is inspired by the text[StSk2005], which was used as a secondary text in my course. Inparticular, the rst half of the course is devoted almost exclusivelyto measure theory on Euclidean spaces Rd(starting with the moreelementary Jordan-Riemann-Darboux theory, and only then movingon to the more sophisticated Lebesgue theory), deferring the abstractaspects of measure theory to the second half of the course. I foundixx Prefacethat this approach strengthened the students intuition in the earlystages of the course, and helped provide motivation for more abstractconstructions, such as Caratheodorys general construction of a mea-sure from an outer measure.Most of the material here is self-contained, assuming only anundergraduate knowledge in real analysis (and in particular, on theHeine-Borel theorem, which we will use as the foundation for ourconstruction of Lebesgue measure); a secondary real analysis text canbe used in conjunction with this one, but it is not strictly necessary.A small number of exercises however will require some knowledge ofpoint-set topology or of set-theoretic concepts such as cardinals andordinals.A large number of exercises are interspersed throughout the text,and it is intended that the reader perform a signicant fraction ofthese exercises while going through the text. Indeed, many of the keyresults and examples in the subject will in fact be presented throughthe exercises. In my own course, I used the exercises as the basisfor the examination questions, and signalled this well in advance, toencourage the students to attempt as many of the exercises as theycould as preparation for the exams.The core material is contained in Chapter 1, and already com-prises a full quarters worth of material. Section 2.1 is a much moreinformal section than the rest of the book, focusing on describingproblem solving strategies, either specic to real analysis exercises, ormore generally applicable to a wider set of mathematical problems;this section evolved from various discussions with students through-out the course. The remaining three sections in Chapter 2 are op-tional topics, which require understanding of most of the material inChapter 1 as a prerequisite (although Section 2.3 can be read aftercompleting Section 1.4.NotationFor reasons of space, we will not be able to dene every single math-ematical term that we use in this book. If a term is italicised forreasons other than emphasis or for denition, then it denotes a stan-dard mathematical object, result, or concept, which can be easilyNotation xilooked up in any number of references. (In the blog version of thebook, many of these terms were linked to their Wikipedia pages, orother on-line reference pages.)Given a subset E of a space X, the indicator function 1E : X Ris dened by setting 1E(x) equal to 1 for x E and equal to 0 forx , E.For any natural number d, we refer to the vector space Rd:=(x1, . . . , xd) : x1, . . . , xd R as (d-dimensional) Euclidean space.A vector (x1, . . . , xd) in Rdhas length[(x1, . . . , xd)[ := (x21 +. . . +x2d)1/2and two vectors (x1, . . . , xd), (y1, . . . , yd) have dot product(x1, . . . , xd) (y1, . . . , yd) := x1y1 +. . . +xdyd.The extended non-negative real axis [0, +] is the non-negativereal axis [0, +) := x R : x 0 with an additional elementadjointed to it, which we label +; we will need to work with thissystem because many sets (e.g. Rd) will have innite measure. Ofcourse, +is not a real number, but we think of it as an extended realnumber. We extend the addition, multiplication, and order structureson [0, +) to [0, +] by declaring++x = x + + = +for all x [0, +],+ x = x + = +for all non-zero x (0, +],+ 0 = 0 + = 0,andx < + for all x [0, +).Most of the laws of algebra for addition, multiplication, and ordercontinue to hold in this extended number system; for instance ad-dition and multiplication are commutative and associative, with thelatter distributing over the former, and an order relation x y ispreserved under addition or multiplication of both sides of that re-lation by the same quantity. However, we caution that the laws ofxii Prefacecancellation do not apply once some of the variables are allowed to beinnite; for instance, we cannot deduce x = y from ++x = ++yor from + x = + y. This is related to the fact that the forms+ + and +/ + are indeterminate (one cannot assign avalue to them without breaking a lot of the rules of algebra). A gen-eral rule of thumb is that if one wishes to use cancellation (or proxiesfor cancellation, such as subtraction or division), this is only safe ifone can guarantee that all quantities involved are nite (and in thecase of multiplicative cancellation, the quantity being cancelled alsoneeds to be non-zero, of course). However, as long as one avoids us-ing cancellation and works exclusively with non-negative quantities,there is little danger in working in the extended real number system.We note also that once one adopts the convention + 0 =0 + = 0, then multiplication becomes upward continuous (in thesense that whenever xn [0, +] increases to x [0, +], andyn [0, +] increases to y [0, +], then xnyn increases to xy)but not downward continuous (e.g. 1/n 0 but 1/n + , 0 +). This asymmetry will ultimately cause us to dene integrationfrom below rather than from above, which leads to other asymmetries(e.g. the monotone convergence theorem (Theorem 1.4.44) appliesfor monotone increasing functions, but not necessarily for monotonedecreasing ones).Remark 0.0.1. Note that there is a tradeo here: if one wantsto keep as many useful laws of algebra as one can, then one canadd in innity, or have negative numbers, but it is dicult to haveboth at the same time. Because of this tradeo, we will see twooverlapping types of measure and integration theory: the non-negativetheory, which involves quantities taking values in [0, +], and theabsolutely integrable theory, which involves quantities taking values in(, +) or C. For instance, the fundamental convergence theoremfor the former theory is the monotone convergence theorem (Theorem1.4.44), while the fundamental convergence theorem for the latter isthe dominated convergence theorem (Theorem 1.4.49). Both branchesof the theory are important, and both will be covered in later notes.One important feature of the extended nonnegative real axis isthat all sums are convergent: given any sequence x1, x2, . . . [0, +],Notation xiiiwe can always form the sum

n=1xn [0, +]as the limit of the partial sums

Nn=1xn, which may be either niteor innite. An equivalent denition of this innite sum is as thesupremum of all nite subsums:

n=1xn = supFN,F nite

nFxn.Motivated by this, given any collection (x)A of numbers x [0, +] indexed by an arbitrary set A (nite or innite, countable oruncountable), we can dene the sum

Ax by the formula(0.1)

Ax = supFA,F nite

Fx.Note