CONVERGENCE OF PROBABILITY MEASURES REVISITED *

Gabriella Salinetti

Universita “La Sapienza” 00185 Roma, Italy

Roger J-B Wets

University of California-Davis, CA 95616

Abstract. The hypo-convergence of upper semicontinuous functions pro-vides a natural framework for the study of the convergence of probabilitymeasures. This approach also yields some further characterizations of weakconvergence and equi-tightness.

Keywords: Narrow (weak, weak*) convergence, epi-convergenceDate: April 2, 1987Printed: July 12, 2005

* This research was supported in part by MPI, Projects: “Calcolo Stocastico eSistemi Dinamici Statistici” and “Modelli Probabilistic” 1984, and by the NationalScience Foundation.

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1. ABOUT CONTINUITY AND MEASURABILITY

A probabilistic structure —a space of possible events, a sigma-field of (observable) subcollections of

events, and a probability measure defined on this sigma-field— does not have a built-in topological

structure. This is the source of many technical difficulties in the development of Probability

Theory, in particular in the theory of stochastic processes. Much progress was made, in reconciling

the measure-theoretic and topological viewpoints, by the study of limits in terms of the weak*-

convergence of probability measures, also called weak convergence [6], [17]. In this paper, we

approach these questions from a fundamentally different point of view, although eventually we

show that weak*-convergence and convergence in the sense introduced here, coincide for probability

measures defined on separable metric spaces. We proceed by a “direct” construction: it is shown

that the spaces of probability measures is in one-to-one correspondence with a certain space of

upper semicontinuous functions, called sc-measures, for which there is a natural topology, and

thus an associated notion of convergence. This means that instead of relying on the pre-dual to

generate the notion of convergence, we use the “topological” properties of the space of probability

sc-measures itself, and much insight is gained by doing so. An important consequence of this

approach is that it allows us to identify, in topological terms, a number of properties of spaces of

probability measures in terms of a topological structure on the underlying probability space.

The major tool is the theory of epi- or hypo-convergence that has been developed in Opti-

mization Theory to study the limits of (infinite-valued) semicontinuous functions. Functions are

said to hypo-converge if their hypographs converge (as sets); the hypograph of an extended-real

valued function consists of all points on and below its graph. This “global” view of functions pro-

vided by the hypographical approach is particularly appealing when dealing with limit theorems

in Probability Theory. Hypo-convergence is not blinded by what happens at single points, a cause

of major chagrin when working with standard pointwise convergence, but takes into account the

behavior of the (converging) functions in the neighborhood of each single point.

We associate to every probability measure, its restriction to the hyperspace of a given class

of closed sets. It is easy, but crucial, to observe that the resulting function, called an sc-measure,

is upper semicontinuous with respect a certain topology on this (hyper)space of closed sets. Now,

limits can be defined in terms of the hypo-limits of these sc-measures as is done in Section 4. The

structural compactness of the space of upper semicontinuous functions with the hypo-topology is

the key to a number of limit results, in particular in the study of the role played by equi-tightness,

cf. Sections 4 and 5. In Section 3, we show that when the probability measures are defined on

the Borel sigma-field of a metric space, hypo-limits of probability sc-measures and weak*-limits

of the associated sequences of probability measures coincide, providing us with an indirect proof

of a Riesz-type representation theorem for the space of sc-measures. We think that this new

characterization of weak*-convergence, that supplements those that have already been studied

extensively [14] and which at least conceptually is non-standard, is by its intrinsic nature closely

related to the usual notion of convergence of the distribution functions of random vectors. In

fact, one may feel that it would be much more natural to approach convergence in distribution

for random vectors in terms of the hypo-convergence of their distribution functions rather than

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through the equivalent but less meaningful notion of pointwise convergence on the continuity set.

The reader can convince himself of this, all what is needed is the definition of hypo-convergence

[10, Section I] and the accompanying geometric interpretation.

We work in general with probability measures defined on a separable metric space, but we

reserve a special place to the case when in addition the underlying space is compact. For this, there

are some technical and didactic reasons —namely in the compact case the notion of convergence

of sets that we introduce corresponds to the usual notion of set-convergence— but in addition the

compact case played an important role in our research on the convergence of stochastic infima

(extremal processes) that originally motivated this work. This was first elaborated in the presen-

tation of these results in a communication to the European Meeting of Statisticians at Palermo

(September 1982), and also at the Third Course in Optimization Theory and Related Fields at

Erice (September 1984). The connections to problems in stochastic homogenization and related

questions in the Calculus of Variations was brought to our attention by a recent article of De Giorgi

[9], whose research is following a path that in some ways, is parallel to ours.

To assume that the domain of definition of the probability measures is compact, or even

locally compact, is not a standard assumption in probability theory, it would exclude a large

number of functional spaces that are of interest in the study of stochastic processes. However

that is only true if we restrict ourselves to a “classical” functional view of stochastic processes.

Instead, if we approach the theory of stochastic processes as in [19, (Sections 3 or 6), 20] where

paths are viewed as elements of a space of semicontinuous functions which we equip with the

epi-topology, or when paths are identified with their graphs and the space of graphs is given the

topology of set-convergence, then the underlying space is actually compact. Specific examples

of such constructions can be found in [19, 21] and in the work of Dal Maso and Modica [8] on

the stochastic homogenization problem (the space of integral functionals on Lploc(R

n) is given the

epi-topology).

Because set-convergence —or the variant that we introduce here to handle the infinite dimen-

sional case, is not yet a familiar tool in probabilistic circles— in the appendices the proofs are

given in painstaking detail. The alerted reader can of course skip much or all of this, and devote

her or his full attention to the actual substance of the results.

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2. CONVERGENCE OF SETS AND SEMICONTINUOUS FUNCTIONS

Let (E, d) be a separable metric space with τ the topology generated by the metric d. By F , or

F(E), we denote the hyperspace of τ -closed subsets of E. We endow F with the topology T that

corresponds to the following notion of convergence: for {F ;F n, n ∈ lN}

F = T - limn→∞

F n

if for all x ∈ E,

d(x, F ) = limn→∞

d(x, F n);

i.e., T is generated by the pointwise convergence of the distance functions {d(·, F n), n ∈ lN} to

d(·, F ). For any nonempty set D ⊂ E,

d(x,D) := infy∈D

d(x, y)

and

d(x, ∅) := ∞.

Francaviglia, Lechicki and Levi have studied extensively the properties of T and related topologies

and uniformities [11], [15]1. In particular, they point out that T -convergence can be characterized

in the following terms. For any η > 0, and nonempty set D in E, the open η-fattening of D is the

set

ηoD := {x ∈ E∣

∣ d(x,D) < η},

with ηD := clτηoD the η-fattening of D. By definition, for some fixed z in E, usually 0 if E is a

linear space, we set

ηo∅ := {x ∈ E∣

∣ d(x, z) > η−1}.

We also reserve the notations

Boη(x) :== {z

∣

∣ d(x, z) < η}, and Bη(x) = {z∣

∣ d(x, z) ≤ η},

for the open and closed balls of radius η and center x.

Proposition 2.1. [11, Propositions 2.1 and 2.2] Suppose {F ;F n, n ∈ lN} ⊂ F . Then F =

T - limn→∞ F n if and only if

(i) for all x ∈ E, to every pair 0 < ε < η, there corresponds n′ such that for all n ≥ n′,

F ∩ Boε (x) 6= ∅ ⇒ F n ∩ Bo

η(x) 6= ∅; (2.1)

(ii) for all x ∈ E, to every pair 0 < ε < η, there corresponds n′ such that for all n ≥ n′,

F ∩ Bη(x) = ∅ ⇒ F n ∩ Bε(x) = ∅. (2.2)

1 They call T the Wijsman topology, but that does not seem to be totally appropriate, since

already Choquet in his 1947 paper “Convergences” (Annales de l’Univ. de Grenoble, 23) introduces

this notion for the convergence of sets.

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It is not difficult to see that condition (i) can be simplified to read: for all x ∈ E and every

0 < ε, there exists n′ such that for all n ≥ n′,

F ∩ Boε (x) 6= ∅ ⇒ F n ∩ Bo

ε (x) 6= ∅;

It follows immediately from this proposition that T - limn→∞ F n = ∅ if and only if to any bounded

set Q, there corresponds n′ such that for all n ≤ n′,

F n ∩ Q = ∅; (2.3)

if d is a bounded metric, this means that the F n are necessarily empty when n is sufficiently large.

This notion of convergence for closed sets is related to, but is more restrictive than, the more

standard definition of set-convergence, by which one means that

F = limn→∞

F n (2.4)

if

lim supn→∞

F n ⊂ F ⊂ lim infn→∞

F n (2.5)

where

lim supn→∞

F n : ={

τ -cluster points of all sequences{xn}∞1∣

∣xn ∈ F n}

(2.6)

= ∩H∈Hτ - cl(∪n∈HF n), (2.7)

and

lim infn→∞

F n : ={

τ -limits of all sequences{xn}∞1∣

∣xn ∈ F n}

(2.8)

= ∩H∈H#τ - cl(∪n∈HF n), (2.9)

here H is the (Frechet) filter on lN = {1, 2, . . .} and H# its grill, i.e.

H# = {H ⊂ lN∣

∣H ∩ H ′ 6= ∅,∀H ′ ∈ H}.

The grill H# consists of all infinite (countable) subsets of lN. Since H ⊂ H#, we always have that

lim infn→∞

F n ⊂ lim supn→∞

F n, (2.10)

and thus F = lim F n actually means that equality holds in (2.5). From these definitions, we

immediately have the following proposition; the details can be found in Appendix A.

Proposition 2.2. [11, Prop. 2.3, Thm. 2.6] Consider {F n, n = 1, . . .} a sequence in F(E). Then

F = T - limn→∞ F n implies that F = limn→∞ F n. The converse also holds if (E, d) is boundedly

compact (i.e., every closed ball is compact).

As a simple example of a case when the converse does not hold, take E = `1, F n = the unit

vector on the nth axis. Then lim F n = ∅ whereas T - lim F ndoes not exist. (Another example:

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E = (0, 1) with F n = n−1.) Some further properties of T and set-convergence are reviewed

in Appendix A; note that a space that is boundedly compact always is locally compact but not

conversely [5].

Francaviglia, Lechicki and Levy prove that (F , T ) is metrizable and separable [11, Theorem

4.2] which of course also implies that it is regular and has a countable base, since singletons are

closed. It is these latter properties that play a key role in what follows. (One can also rely on

the separability of (E, τ), and the characterization of convergence provided by Proposition 2.1 to

obtain a countable base.) For easy reference we record these facts in the next theorem. The second

assertion is well documented in the literature [12], [15], [9, Proposition 3.2].

Theorem 2.3. Suppose (E, d) is separable. The topological space (F(E), T ) has a countable base

and is regular (which also means that it is separable and metrizable, since it is T1). Moreover it is

compact, if in addition (E, d) is boundedly compact.

We note that there is no loss of generality in introducing T -convergence in terms of sequences

instead of nets (or filters).

Now let us consider SCu(F ; [0, 1]), the space of T -upper semicontinuous functions (u.sc.) on

F with values in [0, 1]. Recall that a function υ(F ) → [0, 1] is T -u.sc. at F whenever

υ(F ) ≥ lim supn→∞

υ(F n)

for all sequences {F n}∞n=1 with T - limn→∞ F n = F .

Next, we endow SCu(F ; [0, 1]) with a convergence structure based on the sequential conver-

gence of the hypographs. Recall that the hypograph of υ : F → [0, 1] is the set

hypoυ := {(F, α) ∈ (F × R)∣

∣ υ(F ) ≥ α}, (2.11)

i.e., all points that lie on or below the graph of υ.

Definition 2.4. A sequence of functions {υn, n ∈ lN} in SCu(F ; [0, 1]) hypo-converges to υ ∈ SCu

at F ∈ F if

(i) whenever F = T - limn→∞ F n, then

lim supn→∞

υn(F n) ≤ υ(F ), (2.12)

and

(ii) for some sequence {F n}∞n=1 with F = T - limn→∞ F n,

lim infn→∞

υn(F n) ≥ υ(F ). (2.13)

Note that the first condition could equivalently be formulated as follows: for any subsequence

H ∈ H#, and collection {F n, n ∈ H} with F = T - limn∈H F n, (2.12) holds when ngoes to ∞ on

H. This observation is also useful in showing [10, Proposition 1.9] that hypo-convergence (at all

F in F) corresponds to the (sequential) set-convergence of the hypographs, i.e.,

υ = hypo - limn

υn if and only if hypoυ = limn→∞

hypoυn. (2.14)

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If E is a compact metric space, hypo-convergence of bounded upper semicontinuous real valued

functions can be characterized in terms of convergence of the Hausdorff distances between their

graphs [4].

Of crucial importance to the ensuing development is the following compactness result. This

theorem, an application of a general result about convergence of sets to the space of hypographs,

has a long history that starts with a result of Zoretti [24] in the complex plane; Hausdorff, Lubben,

Urysohn, Blaschke and Marczewski, having contributed in bringing the theorem in its present form.

In Appendix B, we give a direct proof patterned after the argument used in [13] for sequences of

sets (see also [23] for sequences of functions). A very elegant proof appears in [2, Theorem 2.22]

that relies directly on the definition of hypo-convergence.

Theorem 2.5. Suppose (E, d) is a separable metric space, and F is the hyperspace of closed

subsets of E, that we equip with the topology T of pointwise convergence of the distance functions.

Then (SCu(F ; [0, 1],hypo) is sequentially compact, i.e., any sequence {υn ∈ SCu(F ; [0, 1]), n ∈ lN}

contains a subsequence that hypo-converges to a function υ in SCu(F ; [0, 1]). If, in addition (E, d)

is boundedly compact, then (SCu(F ; [0, 1]),hypo) is a regular compact topological space.

3. SC-MEASURES AND SC-PREMEASURES ON F(E)

An sc-premeasure λ is a (set-)function on F(E) with the following properties:

(i) nonnegativity: λ(F ) ≥ 0 for all F ∈ F ;

(ii) increasing (nondecreasing): if for any F1, F2 in F ,

λ(F1) ≤ λ(F2) whenever F1 ⊂ F2;

(iii) λ is T -u.sc. (upper semicontinuous) on F .

It is strongly superadditive if for an F1, F2, in F ,

λ(F1) + λ(F2) ≤ λ(F1 ∪ F2) + λ(F1 ∩ F2). (3.1)

If, it actually is modular [22], i.e., for any F1, F2 in F ,

λ(F1) + λ(F2) = λ(F1 ∪ F2) + λ(F1 ∩ F2), (3.2)

then λ is an sc-measure. It is a probability sc-measure if in addition

λ(∅) = 0, λ(E) = 1. (3.3)

It is said to be tight if sup{λ(K)∣

∣K ⊂ E, compact } = 1.

Sc-measures (semicontinuous measures) defined on F(E) are of course intimately related to

measures defined on B(E), the Borel field generated by F , the τ -closed subsets of E. Conceptually,

however, there is a basic difference between measures and sc-measures. The measure-calculus relies

on the underlying sigma-field structure (countable additivity, etc.), the calculus of sc-measures is

topological in nature, and consequently provides a richer structure for studying convergence, and

other limit questions. It is, however, possible to identify probability measures and sc-measures as

we show next. The results are of the same nature as Kisynki’s Theorem and related results of

Topsøe [22].

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Theorem 3.1. There is a one-to-one correspondence between tight probability measures on B(E)

and tight probability sc-measures on F(E). More precisely, given a probability measure µ on

B(E), then the restriction of µ to F(E) is a probability sc-measure. And given λ, a probability

sc-measure, there exists a unique probability measure µ on B(E) such that µ = λ on F(E). The

tightness condition is, of course, superfluous if the metric space (E, τ) is complete.

Proof. Suppose µ is a probability measure on B and λ is its restriction to F . Clearly, it is enough

to check if λ is T -u.sc. Since T has a countable base, it suffices to show that

lim supn→∞

λ(F n) ≤ λ(F )

whenever F = T - lim F n. First observe that

F = lim supn→∞

F n = ∩H∈H cl(∪n∈HF n) ⊃ ∩H∈H ∪n∈H F n := F ′

where the first two identities follow from Proposition 2.2, and (2.7). Since µ is a probability

measure and λ(F n) = µ(F n) we have that

lim supn→∞

λ(F n) = lim supn→∞

µ(F n) ≤ µ(F ′) ≤ µ(F ) = λ(F ).

Of course, λ will be tight whenever µ is tight.

If λ is a probability sc-measure on F , we set µ = λ on F and define for every open set

G,µ(G) = 1 − λ(E \ G). We see that µ is an increasing modular set-function on A, the field

consisting of finite unions of open and closed sets. We can now appeal to the standard argument

to extend µ to a probability measure on B, the sigma-field generated by A [1, Theorem 1.3.10].

The tightness of λ guarantees not only the tightness of µ, but also that µ is a probability measure.

This extension is unique. In fact, since (E, τ) is a separable metric space, for every A ∈ B, we have

that µ(A) = sup{λ(F )∣

∣F ⊂ A}.

As can be expected from the preceding theorem, measures and sc-measures have many common

properties. Of immediate interest are certain continuity properties used in the sequel. By bdry D

we denote the τ -boundary of a set D ⊂ E.

Proposition 3.2. Suppose λ : F → [0, 1] is a sc-premeasure. Then given any nonempty F ∈ F ,

the set function ε 7→ λ(εF ) : R+ → [0, 1] has at most countably many discontinuity points and

λ(F ) = limε↓0 λ(εF ). Thus, if λ is a probability sc-measure, the family of sets {εF∣

∣ ε ≥ 0} contains

at most countably many sets such that λ(bdry εF ) > 0.

Proof. First note that F = T - limε↓0 εF , see A.5. Since λ is T -u.sc at F and increasing, and hence

λ(F ) ≤ lim infε↓0 λ(εF ), it follows that λ(F ) = limε↓0 λ(εF ). Note that this argument also implies

that ε 7→ λ(εF ) is upper semicontinuous on R+; if ε1 > 0 and ε ↓ ε1 simply apply the preceding

argument to ε1F ∈ F and ε − ε1 ↓ 0. The assertion of at most countable discontinuity points

follows directly from the (topological) argument given in [7, iv p.4] that applies to all monotone

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functions. Finally, if λ is a probability sc-measure, for any ε1 > 0 with λ(bdry ε1F ) > 0, we have

for any ε′ < ε1

λ(ε′F ) + λ(bdry ε1F ) = λ(ε′F ∪ bdry ε1F ) + λ(∅) ≤ λ(ε1F ).

Taking all of the above into account, yields

limε′↓ε1

λ(ε′F ) = λ(ε1F ) ≥ λ(bdry ε1F ) + limε′↑ε1λ(ε′F )

which shows that ε1 is a discontinuity point of ε 7→ λ(εF ) and there are at most a countable

number of such points.

The correspondence between probability measures and sc-measures carries over to the nat-

ural convergence notions for probability measures (weak convergence) and sc-measures (hypo-

convergence). In those terms it is a “bicontinuous” correspondence, as is demonstrated next.

Recall that weak convergence, or more precisely weak* convergence, of a sequence of probabil-

ity measures {µn : B → [0, 1], n ∈ lN} to a probability measure µ : B(E) → [0, 1], denoted

µ = weak*- limn→∞ µn means that

∫

gdµ = limn→∞

∫

gdµn (3.4)

for any bounded continuous g : E → R, or equivalently [6, Theorem 2.1]

lim supn→∞

µn(F ) ≤ µ(F ) for all closed sets F, (3.5)

lim infn→∞

µn(G) ≥ µ(G) for all open sets G, (3.6)

limn→∞

µn(A) = µ(A) for all sets A ∈ cont µ, (3.7)

where

cont µ := {A ∈ B∣

∣ µ(bdryA) = 0}. (3.8)

Theorem 3.3. Suppose (E, τ) is a separable metric space, and {µ;µn, n ∈ lN} is a family of

tight probability measures on B(E), and {λ;λn, n ∈ lN} the corresponding family of probability

sc-measures on F(E). Then

µ = weak*- limn→∞

µn,

if and only if

λ = hypo-limn→∞

λn.

Proof. If λ = hypo-limn→∞

λn then (3.5) follows directly from (2.12), since µn = λn on F , and hence

µ = weak*- lim µn.

To prove the converse, let µ = weak*- limn→∞ µn and let F = T - limn→∞ F n. Let An =

cl∪m≥nF m. The An are monotone decreasing and converge to F . But F n ⊂ An. Hence, for any

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fixed m, lim supn µn(F n) ≤ lim supn µn(An) ≤ µ(Am). The proof is completed by observing that

this holds for all m and that µ(Am) tends to µ(F ) when you let m go to ∞.

Let {εn, n ∈ lN} be a strictly positive sequence converging monotonically to 0. In view of

A.5, for any closed set F , the sequence of sets εnF T -converges to F . For any fixed η > 0, let

{ηn, n ∈ lN} be the sequence where ηn = εn if εn ≥ η, and ηn = η otherwise. The sequence ηnF

converges to ηF , see again A.5. Because of weak*-convergence of the probability measures, for

every η > 0,

lim infn→∞

µn(ηnF ) ≥ lim infn→∞

µn(ηonF ) ≥ µ(ηoF ) ≥ µ(F ).

Because this holds for every η > 0, it follows that lim infn µn(εnF ) ≥ µ(F ).

Theorem 3.3 provides a new characterization of the weak (narrow) convergence of probability

measures. The classical results of Prohorov [6, Section 6] could be obtained as a direct consequence

of this characterization, and the compactness results of Section 2. However, to demonstrate that

this approach (via sc-measures) does not need to rely on the standard theory, we provide in

Appendix B an independent derivation of Prohorov’s Theorems that highlights the relationship

between equi-tightness and the semicontinuity of sc-measure at ∅.

4. TIGHTNESS AND EQUI-SEMICONTINUITY

Theorem 2.5 can be viewed as a generalization of a version of Helly’s Theorem for probability

sc-measures. The standard formulation of Helly’s Theorem is, however, in terms of a “pointwise”

convergence of the distribution. From the results of this section it will follow that hypo-convergence

of sc-measures can be given a pointwise characterization, which also leads us to relate equi-tightness

to an equi-upper semicontinuity condition “at infinity.”

The relationship between pointwise and hypo-convergence has been studied in [10]. Neither

one implies the other, unless the collection of functions is equi-u.sc. [10, Theorem 2.9]. A family

U ⊂ SCu(F ; [0, 1]) is equi-upper semicontinuous at F (with respect to the T -topology) if to every

ε > 0 there corresponds a T -neighborhood V of F such that for all υ ∈ U ,

supV ∈V

υ(V ) ≤ υ(F ) + ε. (4.1)

Proposition 4.1. Suppose E is a separable metric space and {λ;λn, n ∈ lN} are equi-tight prob-

ability sc-measures such that λ = hypo-limn→∞

λn. Then

(i) the sequence is equi-u.sc. on cont λ,

(ii) ∅ ∈ cont λ. where cont λ = {F ∈ F∣

∣ λ(bdryF ) = 0}.

Proof. From Theorem 3.3, (3.7) and Theorem 3.1 follows that for all F ∈ cont λ, limn→∞ λn(F ) =

λ(F ). This means that on cont λ we have both hypo- and pointwise convergence, and this only

occurs if the sequence is equi-u.sc. on cont λ [10, Theorem 2.18].

Part (ii) follows from: bdry ∅ = ∅ and λ(∅) = 0.

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Theorem 4.2. Suppose E is a separable metric space and {λ;λn, n ∈ lN} are probability sc-

measures. Then λ = hypo-limn→∞

λn if and only if λ(F ) = limn→∞ λn(F ) for all F ∈ cont λ.

Proof. For the “only if” part, see above. For the converse we rely on Proposition 3.2 and the fact

that λ is T -u.sc. Indeed, they imply that given any ε1 > 0, there always exists ε ∈ (0, ε1) such

that εF ∈ cont λ and λ(εF ) < λ(F ) + ε1. Because of pointwise convergence on cont λ, we have

that

lim supn→∞

λn(F ) ≤ limn→∞

λn(εF ) = λ(εF ) < λ(F ) + ε1,

whatever be ε1 > 0. Hence lim supn→∞ λn(F ) ≤ λ(F ) for all F ∈ F . But this via Theorem 3.1,

(3.5), and Theorem 3.3 yields the hypo-convergence of the λn to λ.

If we are given a sequence {λn, n ∈ lN} of probability sc-measures that hypo-converges to λ

and it also pointwise converges on cont λ, this would not yet imply that λ is a probability sc-

measure: modularity and λ(∅) = 0 might still fail to be satisfied. If (E, τ) is boundedly compact

however, it does suffice to have ∅ ∈ cont λ, as follows from the next proposition.

Proposition 4.3. Suppose (E, τ) is a boundedly compact separable metric space, and {λn, n ∈ lN}

are probability sc-measures on F(E). Then the following statements are equivalent:

(i) the sequence {λn, n ∈ lN} is equi-tight;

(ii) the sequence {λn, n ∈ lN} is equi-u.sc. at ∅;

(iii) any sequence {F n, n ∈ lN} with ∅ = limn→∞ F n implies lim supn→∞ λn(F n) = 0.

Proof. Using the characterization of basic neighborhood systems of ∅, and recalling that the closed

balls in (E, τ) are compact, we see that the {λn, n ∈ lN} are equi-u.sc. at ∅ if and only if to every

ε > 0, there corresponds a compact ball Kε such that for all n

λn(F ) ≤ λn(∅) + ε = ε whenever F ∩ Kε = ∅,

or equivalently λn(Kε) > 1 − ε. In other words, if and only if the sequence is equi-tight.

That (i) implies (iii) follows from Theorem B.3 and Proposition 4.1. For the converse, if (i)

or equivalently (ii) does not hold, by definition of equi-usc, then there exist ε > 0 and a sequence

{Fn, n ∈ lN} that converges to ∅ so that for all n, λn(Fn) > λn(∅) + ε = ε. But that means that

also (iii) could not hold.

Acknowledgment We are grateful to Professor G. Beer for a substantial number of sugges-

tions and pointers, in particular in connection with the boundedly compact assumption in Propo-

sition 2.2, and its relationship to local compactness. Also for pointing out, as well as Professor W.

Vervaat, the need for the tightness condition in Theorem 3.1.

11

APPENDIX A

This is a collection of facts about set- and T -convergence that are used in the text. It is always

assumed that (E, d) is a separable metric space.

A.1. T -convergence implies set-convergence

Proof. To being with we show that if F = T - lim F n, then F ⊂ lim infn→∞ F n. There is nothing

to prove if F = ∅, so let us assume that F is nonempty. By Proposition 2.1, if x ∈ F and ε > 0,

then Boε (x) ∩ F n 6= ∅ for all n sufficiently large. This means that for all H ∈ H#, x ∈ τ - cl(∪nF n)

and hence, by (2.9), x ∈ lim inf F n.

Next, let us show that F ⊃ F ′ := lim supn→∞ F n, or equivalently that E \F ⊂ E \F ′. There

is nothing to prove if F = E, so let us assume E \ F 6= ∅. Suppose x ∈ E \ F , then there exists

η > 0 such that F ∩ Bη(x) = ∅. By Proposition 2.1 this implies that for any ε ∈ (0, η), for n

sufficiently large F n ∩ Bε(x) = ∅. Thus there exists H ∈ H such that x /∈ τ - cl(∪n∈HF n), and by

(2.7) this implies that x /∈ lim sup F n, i.e., x ∈ E \ F ′.

The fact that T -convergence and set convergence are the same when (E, d) is a Euclidean

space is well know, see [18, Theorem 2.2] for example. Beer [4] pointed out that this equivalence

can only be obtained for spaces whose closed balls are compact.

A.2. F1 = lim F n1 , F2 = lim F n

2 implies F1 ∩ F2 = lim[(F1 ∩ F2) ∪ (F n1 ∩ F n

2 )]

Proof. Set F = F1 ∩ F2, Fn = F n

1 ∩ F n2 . If x ∈ lim sup(F ∪ F n) this means that there exists

H ∈ H#, xn ∈ F ∪ F n for all n ∈ H such that x = lim xn. If xn ∈ F infinitely often then

x ∈ F , otherwise x ∈ lim supF n ⊂ lim sup F n1 ∩ lim sup F n

2 = F1 ∩ F2 = F ; the inclusion can

be obtained from the definition (2.6) of lim sup, or one can consult [13, section 25. II]. Hence

lim sup(F ∪ F n) ⊂ F and, since F ⊂ F ∪ F n implies F ⊂ lim inf(F ∪ F n), the assertion follows

from (2.5).

A.3. F1 = T - lim F n1 , F2 = T - lim F n

2 implies F1 ∪ F2 = T - lim(F n1 ∪ F n

2 ).

Proof. Use the definition of T -convergence

A.4. F = lim F n, F ⊂ F n. Then, for all compact sets K, F ∩ K = T - lim F n ∩ K.

Proof. We use the characterization of T -convergence given by Proposition 2.1. For any pair

0 < ε < η, F ∩K ∩Boε(x) 6= ∅ implies that F n ∩K ∩Bo

η(x) 6= ∅ for all n, since F ⊂ F n. Now, if for

some η > 0, F ∩K ∩Bη(x) = ∅, but there exists ε > 0, and H ∈ H# such that F n ∩K ∩Bε(x) = ∅

for all n ∈ H, it would mean that lim supn→∞(F n ∩ K ∩ Bε(x)) is nonempty. But, in view of

lim supn→∞

(F n ∩ K ∩ Bε(x)) ⊂ (lim supn→∞

F n) ∩ (K ∩ Bε(x)) ⊂ F ∩ (K ∩ Bη(x))

this would contradict the assumption that F ∩ (K ∩ Bη(x)) = ∅.

A.5. F = T - limε↓0 εF .

Proof. Since (F , T ) is separable (Theorem 2.3) we only need to consider this limit in terms of

a sequence {εn, n ∈ lN} with lim εn = 0. If F = ∅ then the result is a direct consequence of the

12

definition of ε-fattening of the empty set and the comment that follows Proposition 2.1. When F

is nonempty, then for all η > 0 such that F ∩ Boη(x) 6= ∅ it follows that εnF ∩ Bη(x) 6= ∅ for all n,

since F ⊂ εnF . If F ∩ Bη(x) = ∅ then for any ε ∈ (0, η), for n sufficiently large εnF ∩ Bε(x) = ∅

as follows from the fact that E is normal.

A.6. {Cn, n ∈ lN} is an increasing sequence of closed balls with E = T - lim Cn. Then {cl(E \

Cn), n ∈ lN} is decreasing and ∅ = T - lim cl(E \ Cn).

Proof. The sequence is clearly decreasing. For D any bounded set and n sufficiently large D ⊂

intCnand thus D ∩ cl(E \ Cn) = ∅ which via Proposition 2.1 implies that {cl(E \ Cn), n ∈ lN}

T -converges to ∅.

A.7. F = T - lim F n, or more generally F = lim F n, and K ⊂ E compact. Then for all ε > 0,

there exists nε such that for all n ≥ nε, Fn ∩ K ⊂ εF .

Proof. If F = ∅, then F n ∩ K = ∅ for n sufficiently large (Proposition 2.1), in which case the

inclusion is clearly satisfied. If F is nonempty and F n ∩ K is not included in εF , it means that

there exists H ∈ H# such that for all n ∈ H, F n ∩ K ∩ (E \ εF ) 6= ∅. Passing to a subsequence if

necessary, it means that there exists {xn, n ∈ H} such that xn ∈ F n ∩K ∩ (E \ εF ), and by (2.6)

we then have

lim xn = x ∈ (lim supF n) ∩ K ∩ cl(E \ εF ) = F ∩ cl(E \ εF )∩ K,

which completes the proof.

A.8. F = T - lim F n, ∅ 6= K compact and ε > 0. Then for n sufficiently large F n ∩ εK ⊂ ε′F

where ε′ > 2ε.

Proof. The case F = ∅ is argued as in A.7. Otherwise, for contradiction purposes, suppose that

there exists H ∈ H# such that for all n in H, d(xn, F n \ ε′F ) ≤ ε for some xn ∈ K. Passing to a

subsequence, if necessary, we have that lim xn = x ∈ K∩εF , since K is compact, εF = T - lim εF n

and every xn ∈ εF n. On the other hand d(xn, F ) > ε′ − ε and thus d(x, F ) > ε, again by

T -convergence, contradicting the possibility that x ∈ εF .

13

APPENDIX B: HYPO-LIMITS OF SC-MEASURES AND TIGHTNESS

In this section we are interested in the properties of the limit functions of a sequence of probability

sc-measures. In view of Theorem 2.5 we know that there always will be a limit function, at

least for some subsequence, what cannot be guaranteed is that this limit function is a probability

sc-measure. We begin with a general result about sc-premeasures.

Lemma B.1. The space of sc-premeasures is sequentially compact for the hypo-convergence topol-

ogy. In particular, if {λn, n ∈ lN} is a sequence of sc-premeasures such that λn(E) = 1 for all n,

and λ := hypo-limn→∞

λn, then λ is a sc-premeasure with λ(E) = 1.

Proof. Every sequence of sc-premeasures has a hypo-convergent subsequence (Theorem 2.5). The

limit function is then T -u.sc., and clearly it is nonnegative. We need to show that it is an increasing

function. Suppose λ = hypo-limn→∞

λn, and let us consider any pair F1 ⊂ F2 in F . Since λ is the hypo-

limit of the λn, it follows from (2.13) and (2.12) that there exists a sequence with F1 = T - lim F n1

such that λ(F1) = lim λn(F n1 ). Since F2 = T - limn→∞ F2 ∪ F n

1 , see A.3, it follows that

λ(F2) ≥ lim supn→∞

λn(F2 ∪ F n1 ) ≥ lim sup

n→∞

λn(F n1 ) = λ(F1).

Finally, λ(E) = 1 whenever λn(E) = 1 for all n, since by (2.12)

λ(E) ≥ lim supn→∞

λn(E) = 1,

and λ(E) ≤ lim infn→∞ λn(F n) ≤ 1 whatever be the sequence {F n, n ∈ lN} T -converging to E.

Thus the hypo-limit λ of a sequence of probability sc-measures is a sc-premeasure. Without

making additional assumptions about (E, τ), not much more can be said except for a subadditivity

property:

λ(F1) + λ(F2) ≥ λ(F1 ∪ F2) ∀F1, F2 ∈ F .

In general, λ is neither modular, nor is λ(∅) = 0. As the ensuing development will show these two

properties are not unrelated. Let us begin by giving a necessary and sufficient condition for having

λ(∅) = 0.

Lemma B.2. Suppose {λn, n ∈ lN} is a sequence of probability sc-measures on F(E), and λ =

hypo-limn→∞λn. Then λ(∅) = 0 if and only if to every ε > 0, there corresponds a closed ball Cε

and nε such that for all n ≥ nε,

λn(Cε) > 1 − ε. (B.1)

Proof. Fix any ε > 0. Proposition 2.1 and the definition of hypo-convergence yield the existence

of a sequence {F n, n ∈ lN} such that for all n ≥ nε,

F n ∩ Cε = ∅, λ(∅) = limn→∞

λn(F n).

Since for all n ≥ nε, λn(F n) ≤ 1 − λn(Cε) ≤ ε, it implies that 0 ≤ λ(∅) < ε. This holds for all

ε > 0, which means that λ(∅) = 0.

14

Let us now prove the converse. We argue by contradiction. Suppose λ(∅) = 0, but for some

ε > 0 and every closed ball C, there exists HC ∈ H# such that for all n ∈ HC , λn(C) ≤ 1 − ε.

Since

λn(C) + λn(cl(E \ C) ≥ λn(E) = 1,

it follows that for all n ∈ HC , λn(cl(E \C)) > ε. The fact that λ is the hypo-limit of the λn implies

that for all closed ball C,

ε ≤ lim supn→∞

λn(cl(E \ C)) ≤ λ(cl(E \ C)).

Consider any increasing sequence {Cn, n ∈ lN} of closed balls that T -converges to E. This means

that T - limn→∞ cl(E \ Cn) = ∅, cf. A.6. The preceding inequality and the T -u.sc. of λ (Lemma

B.1) would yield the following contradiction.

0 < ε ≤ lim supn→∞

λ(cl(E \ Cn)) ≤ λ(∅) = 0.

This leads us to the following observations. A collection of probability sc-measures Λ on F

(resp. measures M on B) is said to be equi-tight, if to every ε > 0 there corresponds a compact set

Kε ⊂ E such that for all, but a finite number, of λ in Λ (resp. µ ∈ M)

λ(Kε) > 1 − ε, ( resp. µ(Kε) > 1 − ε). (B.2)

If the closed balls of the metric space E are compact, we can rewrite the assertion of Lemma B.2

as follows: λ(∅) = 0 if and only if the sequence {λn, n ∈ lN} is equi-tight. But, as it turns out,

having λ(∅) = 0 is all what is needed to obtain the modularity of λ when E is locally compact.

To show this, one first proves that in the locally compact case, the hypo-limit of a sequence of

probability sc-measures is always a strongly superadditive sc-premeasure. Next, it is shown that

if in addition this hypo-limit has λ(∅) = 0, then λ is actually modular and hence λ is a probability

sc-measure. This means: if the {λn, n ∈ lN} are probability sc-measures on F(E) with E a locally

compact separable metrizable space, and λ = hypo-limn→∞λn, then λ is a probability sc-measure

if and only if the sequence {λn, n ∈ lN} is equi-tight. This follows from the fact that E locally

compact, separable and metrizable admits a boundedly compact metric [24]. When E is given

this metric, we are in the following situation: T -convergence and set convergence coincide, and

(SCu(F ; [0, 1]),hypo) is compact; details can be found in [20].

The same assertion can be made if E is a Polish space —and this is proved below— but equi-

tightness plays then a double role. As in the locally compact case, it guarantees that there is no

escape of the probability mass “at infinity” but it is also used to generate “relative compactness”

in the space of probability measures. Tightness already enters in the proof that the hypo-limit

is strongly superadditive. It essentially allows us to restrict our attention to compact subsets of

E where T -limits coincide with the standard set-limits, and up to some technical adjustments we

can rely, as in the locally compact case, on the built-in relative compactness of the space of u.sc.

functions (with the hypo-topology).

15

Theorem B.3. Suppose {λn, n ∈ lN} is a sequence of probability sc-measures on F(E) where E

is a separable metric space, and λ = hypo-limn→∞

λn. Then

(i) if the sequence {λn, n ∈ lN} is equi-tight, λ is a tight probability sc-measure;

(ii) if λ is a probability sc-measure and E is a Polish space, the sequence {λn, n ∈ lN} is

equi-tight.

Proof. If the {λn, n ∈ lN} are equi-tight, then λ is a sc-premeasure with λ(E) = 1 (Lemma B.1),

but also λ(∅) = 0 (Lemma B.2). Moreover, λ is tight as a direct consequence of the tightness of the

sequence and the definition of hypo-convergence. To show that λ is strongly superadditive (3.1),

observe that since λ is the hypo-limit of the λn, for any pair F1, F2 in F , there exists sequences

such that T - lim F n1 = F1 and T - limn F n

2 = F2 such that

limn→∞

λn(F n1 ) = λ(F1), and lim

n→∞λn(F n

2 ) = F2

Given ε > 0, let Kεbe a compact set such that for all, but finitely many, n: λn(Kε) > 1− ε. Since

the λn are probability sc-measures:

λn(F n1 ) + λn(F n

2 ) = λn(F n1 ∪ F n

2 ) + λn(F n1 ∩ F n

2 )

≤ λn(F n1 ∪ F n

2 ) + λn((F n1 ∩ F n

2 ) ∪ (F1 ∩ F2))

≤ λn(F n1 ∪ F n

2 ) + λn[((F n1 ∩ F n

2 ) ∪ (F1 ∩ F2)) ∩ Kε] + ε.

Taking lim sup on both sides, using the fact that λ is the hypo-limit of the λn (2.12), since F1∪F2 =

T - limn→∞(F n1 ∪ F n

2 ) by A.3, and

F1 ∩ F2 ∩ Kε = T - lim[((F1 ∩ F2) ∪ (F n1 ∩ F n

2 )) ∩ Kε]

(cf. Proposition 2.2, A.2 and A.4), we obtain

λ(F1) + λ(F2) ≤ λ(F1 ∪ F2) + λ(F1 ∩ F2 ∩ Kε) + ε

≤ λ(F1 ∪ F2) + λ(F1 ∩ F2) + ε,

where for the last inequality we used the fact that λ is increasing since it is sc-premeasure. This

yields (3.1) since ε > 0 is arbitrary. To complete the proof of part (i), there remains only to show

that

λ(F1) + λ(F2) ≥ λ(F1 ∪ F2) + λ(F1 ∩ F2). (B.3)

Since λ(∅) = 0, we may as well assume that F1 and F2 are nonempty, the inequality being

trivially satisfied otherwise. Let {Rn, n ∈ lN} and {Sn, n ∈ lN} be sequences of sets such that

T - limn→∞ Rn = F1 ∪ F2, T - limn→∞ Sn = F1 ∩ F2, and

λ(F1 ∪ F2) = limn→∞

λn(Rn), λ(F1 ∩ F2) = limn→∞

λn(Sn).

The existence of such sequences follows as before from (2.13) and (2.12). Pick any η > 0. With

F ′ = cl(E \ ηF2), and using the fact that the λn are probability sc-measures, we have

λn(Rn) + λn(Sn) ≤ λn(Rn ∩ ηF2) + λn(Rn ∩ F ′) + λn(Sn)

≤ λn(ηF2) + λn((Rn ∩ F ′) ∪ Sn) + λn(Rn ∩ F ′ ∩ Sn)

≤ λn(ηF2) + λn((Rn ∩ F ′) ∪ Sn ∪ F1) + λn(Rn ∩ F ′ ∩ Sn).

16

Moreover, the sequence {λn, n ∈ lN} is equi-tight. Let Kε be a compact set such that λn(Kε) >

1 − (ε/2) for n sufficiently large. Thus for n sufficiently large

λn(Rn) + λn(Sn) ≤ λn(ηF2) + λn[((Rn ∩ F ′) ∪ SncupF1) ∩ Kε]

+ λn(Rn ∩ F ′ ∩ Sn ∩ Kε) + ε.

Taking lim sup on both sides, using the fact that λ is the hypo-limit of the {λn, n ∈ lN}, and since

T - lim[((Rn ∩ F ′) ∪ Sn ∪ F1) ∩ Kε] = F1 ∩ Kε

and

∅ = T - limn→∞

(Rn ∩ Sn ∩ F ′ ∩ Kε)

as follows from Proposition 2.2, A.2, A.3 and A.4, we have

λ(F1 ∪ F2) + λ(F1 ∪ F2) ≤ λ(ηF2) + λ(F1 ∩ Kε) + ε,

≤ λ(ηF2) + λ(F1) + ε.

We obtain (B.3) from the above by observing that λ is T -u.sc at F2, T - limn→∞ ηF2 = F2 by A.5,

and that ε > 0 can be chosen arbitrarily small.

To prove the converse, part (ii), observe that since (E, τ) is a Polish space every probability

measure [6, Theorem 1.4], and thus every probability sc-measure (Theorem 3.1) is tight. In par-

ticular, this means that for all ε > 0 there exists K ⊂ E, compact such that λ(K) > 1 − ε. In

turn, this implies that for all η > 0, λ(E \ ηoK) ≤ ε. Since λ = hypo-lim λn,

lim supn→∞

λn(E \ ηoK) ≤ λ(E \ ηoK) ≤ ε

from which it follows that there exists nε such that for all n ≥ nε and η > 0:

λn(ηK) ≥ 1 − λn(E \ ηoK) ≥ 1 − 2ε.

Now, we use the fact that η > 0 is arbitrary, that by A.5, K = T - lim ηK and that the λn are

T -u.sc. at K to conclude that for all n ≥ nε, λn(K) > 1 − 3ε. This means that the sequence

{λn, n ∈ lN} is equi-tight, since there is such a compact set K for any ε > 0.

Proposition 4.3 and Theorem 4.2 allow us, in the compact case, to rephrase Theorem B.3

as follows: Suppose (E, τ) is a compact metric space, {λn, n ∈ lN} is a sequence of probability

sc-measures on F(E) such that λ = limn→∞ λn on cont λ. Then λ is a probability sc-measure if

and only if the {λn, n ∈ lN} are equi - u.sc. at ∅.

The theorems of Prohorov and Varadajaran [6, Section 6], [15, Theorem 6.7] are immediate

consequences of the above. As in [6] or [17], we say that a family M of probability measures is

relatively compact (with respect to the weak* topology on M) if any sequence {µn, n ∈ lN} ⊂ M

contains a subsequence weak converging to a probability measure.

17

Corollary B.4. Prohorov’s Theorems Suppose (E, d) is a separable metric space and M is a

family of probability measures on BE). Then, if M is equi-tight it is relatively compact. Moreover

if (E, d) is a Polish space, then relative compactness of M implies equi-tightness.

Proof. Let Λ = {λ : F(E) → [0, 1]∣

∣ λ = µ on F , µ ∈ M} be the associated family of probability

sc-measures, cf. Theorem 3.1. If M is equi-tight, so is Λ. Moreover, any sequence in Λ contains

a hypo-convergent subsequence (Theorem 2.5) which is necessarily equi-tight, and hence its hypo-

limit is a probability sc-measure (Theorem B.3. (i)). The first assertion now follows directly from

Theorem 3.3.

For the converse, let us first consider the case when M = {µn, n ∈ lN}. It follows directly

from the definition of relative compactness, Theorem 3.3, Theorem B.3. (ii), the definition of equi-

tightness and the fact every probability sc-measure on the Polish space (E, d) is tight, that for all

ε > 0 there exists Kε such that not only µn(Kε) > 1 − ε for all n ∈ lN, but also µ(Kε) > 1 − ε

where µ is any (probability) measure in the weak* closure of M. To complete the proof simply

observe that the weak* topology (on the space of measures on B) is separable, and thus there exists

a dense subset {µn, n ∈ lN} ⊂ M whose weak* closure is equi-tight.

By the way, note that the separability of the weak*-topology is now a direct consequence of

Theorems 3.1 and 2.3.

18

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