Probability Measures on Product Spaces with Uniform Metrics

Discussion Papers of the Max Planck Institute for Research on Collective Goods 2017/6

MAX PLANCK SOCIETY

Martin F. Hellwig

MAX PLANCK SOCIETY Discussion Papers of the MAX PLANCK Max Planck Institute SOCIETY for Research on Collective Goods 2017/6

Probability Measures on Product Spaces with Uniform Metrics

Martin F. Hellwig

May 2017

This version December 2020

Max Planck Institute for Research on Collective Goods, Kurt-Schumacher-Str. 10, D-53113 Bonn https://www.coll.mpg.de Probability Measures on Product Spaces with Uniform Metrics∗

Martin F. Hellwig Max Planck Institute for Research on Collective Goods Kurt Schumacher-Str. 10 D-53113 Bonn, Germany [email protected]

December 23, 2020

Abstract For a countable product of complete separable metric spaces, with a topology induced by a uniform metric, the σ-algebra generated by the open balls, which was introduced by Dudley (1966), coincides with the product σ-algebra. Any probability measure on the product space with this σ-algebra is quasi-separable in the sense that, for any union of open balls that has full measure, there is a countable sub-union that also has full measure. With suitably adapted definitions, the topology of weak convergence on the space of such measures is equivalent to the topology induced by the Prohorov metric. The projection mapping from such measures to sequences of measures on the first ` factors, ` = 1, 2, ..., is a homeomorphism if the range of this mapping is also given a uniform metric. These findings are relevant for the theory of games of incomplete information, where a topology on the space of belief hierarchies that is based on a uniform metric has been proposed as being more appropriate for capturing the continuity properties of strategic behaviour. Key Words: Product spaces with uniform metrics, weak convergence of non-Borel measures, σ-algebras generated by the open balls, quasi-separable measures, Prohorov metric. MSC Classification 60B05 JEL Classification: C02, C72

∗Without implicating them, I thank Eduardo Faingold and Alia Gizatulina for helpful discussions.

1 1 Introduction

Let X1,X2, ... be non-singleton complete separable metric spaces with metrics ρ1, ρ2, ... . Suppose that the product

∞ X = Xk (1.1) k=1 Y has the topology induced by the uniform metric ρu where, for any x and xˆ in X, u ρ (x, xˆ) = sup ρk(πk(x), πk(ˆx)) (1.2) k u and πk is the projection from X to Xk. I use the notation X to indicate u u that X has the topology induced by ρ . Let 0(X ) be the σ-algebra that is u B u generated by the ρ -open balls and let 0(X ) be the space of probability u u Mu u measures on (X , 0(X )). Let CB0(X ) be the space of bounded, ρ - B u continuous, and 0(X )-measurable real-valued functions on X. Say that r B u a sequence µ of measures in 0(X ) converges weakly to a measure u{ } M µ 0(X ) if and only if ∈ M f(x)dµr(x) f(x)dµ(x) (1.3) u → u ZX ZX u for all f CB0(X ). ∈ u u This paper shows that, even though X is non-separable and 0(X ) is not a Borel σ-algebra, yet, under the assumption that the cardinalB c of the continuum is not atomlessly measurable, the topology of weak convergence u on 0(X ) is metrizable by a suitably adapted version of the Prohorov metric.M The argument is similar to the argument for Borel measures on pos- sibly non-separable metric spaces.1 However, whereas the latter argument relies on the fact that, under the given assumption about the cardinal of the continuum, any Borel measure on a metric space is concentrated on a separable set,2 I show that, under the same assumption, every measure in u 0(X ) is concentrated on what I call a quasi-separable set, i.e., a set with Mthe property that any covering of the set by open balls contains a countable subcovering. The proof of the latter result uses the fact that X has a product structure and combines ideas from Banach (1930) and Billingsley (1968).

1 See, e.g., Theorem 5, p. 238, in Billingsley (1968). 2 See Theorem III in Marczewski and Sikorski (1948) or Theorem 2, p. 235, in Billingsley (1968).

2 u u The σ-algebra 0(X ), the space 0(X ), and the topology of weak B u M convergence on 0(X ) were introduced by Dudley (1966, 1967) in order to avoid certainM inconveniences associated with the Borel σ-algebra (Xu) that are induced by the non-separability of Xu.3 Dudley was interestedB in the convergence properties of sequences of stochastic processes when the space of sample paths of the processes has the uniform topology. By using the coarser σ-algebra induced by the open balls, he avoided the diffi culties arising from the large size of the Borel σ-algebra. Dudley’s approach provides an alternative to the use of the Skorokhod topology and associated Borel σ-algebra on the space of the sample paths. Dudley did not actually study the topology that is induced by the con- u cept of weak convergence on 0(X ). Given his interest in the convergence properties of sequences ofM stochastic processes, he merely considered the convergence behaviour of sequences of integrals of bounded continuous functions, including upper and lower integrals for functions that are not measurable with respect to the smaller σ-algebra induced by the open balls. In this analysis, he assumed that the limit measure of such a sequence is concentrated on a separable set; this assumption presumes that the limit measure can be extended to the Borel σ-algebra for the uniform topology so that the result of Marczewski and Sikorski could be appealed to.4 Dudley’s approach thus involves an implicit asymmetry between measures that can be so extended and measures that cannot be so extended. For Dudley’s reasearch programme, this asymmetry did not matter because the limit measures in his convergence theorems satisfied extendability condition anyway. In other contexts, the asymmetry is problematic. In the final section of this paper, I discuss recent developments in game theory that have provided the motivation for the research presented here.

u 2 Quasi-Separability of Measures in 0(X ) M Because each the spaces X1,X2,X3, ... has more than one element, the cardinal of X is at least 2ω = c where ω is the cardinal of the natural numbers 3 See also Wichura (1970), Dudley (1978), and Pollard (1979). 4 Pollard (1979) implicitly makes the same assumption. He uses the separability of the support of the limit measure in order to deﬁne a topology that is smaller than the topology of weak convergence on the space of measures on the σ-algebra generated by the open balls, but suﬃ ciently large to provide for weak convergence in a neighbourhood of the limit measure. Wichura (1970) considers the random variables associated with the measures on the σ-algebra generated by the open balls.

3 and c is the cardinal of the continuum. It is also no greater than c because, as a product of complete separable metric spaces, with the product topology, X itself is a complete separable metric space so the cardinal of X is either ﬁnite or ω or c.5 u The cardinal of the σ-algebra 0(X ) is also equal to c. To see this, note that, by a result of Dudley (1967, p.B 449), the claim is true if c is the smallest cardinal of a dense set in Xu. To see that Xu satisﬁes this condition, for 1 2 k = 1, 2,..., let xk, xk be two distinct elements of Xk and consider the set 1 2 Xˆ = x = (x1, x2,...) X xk x , x for all k . Clearly, Xˆ X has the { ∈ | ∈ { k k} } ⊂ cardinal c, and so does any subset of Xˆ, or of X, that is ρu-dense. Throughout the analysis, I impose the following assumption.

Assumption 1 The cardinal c of the continuum is not atomlessly measurable: no set of cardinality c or less admits a nontrivial atomless measure that is deﬁned on all subsets of the set.

u Deﬁnition 2.1 A measure µ 0(X ) is called quasi-separable if, for any family of ρu-open balls that∈ covers M X, there exists a countable subfamily suchG that the union G of the sets in satisﬁes µ(G ) = 1. G∗ ∗ G∗ ∗ u Proposition 2.2 Under Assumption 1, any measure µ 0(X ) is quasi- separable. ∈ M

The proof of Proposition 2.2 makes use of the outer measure that a u measure µ 0(X ) induces on every subset of X. For a given set U X, ∈ M ⊂ 6 the value µ∗(U) of the outer measure µ∗ that is induced by µ is deﬁned as

∞ ∞ u µ∗(U) = inf µ(Bi) s.t. U Bi and Bi 0(X ) for all i. (2.1) ⊂ ⊂ B i=1 i=1 X [ The following preliminary lemma is of interest in its own right.

u u Lemma 2.3 Let µ be a ﬁnite measure on (X , 0(X )). Let I be a set of u B indices and let Bi i I be a family of open ρ -spheres such that µ(Bi) = 0 for all i I. Then{ } the∈ set ∈ UI := Bi (2.2) i I [∈ has the outer measure µ∗(UI ) = 0.

5 Theorem 13.1.1 in Dudley (2002). 6 See Dudley (2002), p. 89.

4 Proof. For any i I and any natural number k, let ∈ k k k+1 k+2 Π (Bi) := π (Bi) X X ... (2.3) × × × k where π (Bi) is the projection of Bi to the finite product X1 ... Xk. For any set Iˆ I and any k, let × × ⊂ U k := Πk(B ). (2.4) Iˆ i i Iˆ [∈ k For any i and k, π (Bi) is a finite product of open spheres. Therefore, k Π (Bi) is an open set in the product topology on X. For any Iˆ I, therefore, U k is also an open set in the product topology on X . Because⊂ the Iˆ u product σ-algebra on X is coextensive with the σ-algebra 0(X ), it follows that µ(U k) is well defined for all Iˆ I and all k. For anyBk, therefore, the Iˆ formula ⊂ νk(Iˆ) := µ(U k) (2.5) Iˆ defines a set function on the subsets of I. k From (2.3), one finds that, for any i, the sequence Π (Bi) is non- { }k∞=1 increasing. For any Iˆ I, therefore, the sequence U k is also nonde- ⊂ { Iˆ }k∞=1 creasing. For any Iˆ I, therefore, the limit ⊂ k ν∞(Iˆ) := lim ν (Iˆ) (2.6) k →∞ is well defined. I will show that the set function ν∞ is atomless as well as countably additive. I first show that ν∞ is atomless, i.e., that

ν∞( i ) = 0 (2.7) { } for all i I. For any i I, obviously, ∈ ∈

∞ k Π (Bi) = Bi, (2.8) k=1 \ and therefore

k k ν∞( i ) := lim ν ( i ) = lim µ(Π (Bi)) = µ(Bi), (2.9) { } k { } k →∞ →∞ so (2.7) follows from the assumption that µ(Bi) = 0 for all i I. ∈

5 I next show that ν∞ is countably additive. Let Ij, j = 1, 2, ..., be a sequence of disjoint subsets of I. For any k, obviously,

k ∞ ∞ k ∞ k ∞ k ν Ij = µ U = µ(U ) = ν (Ij). (2.10)    Ij  Ij j=1 j=1 j=1 j=1 [ [ X X     The desired result,

∞ ∞ ν∞ Ij = ν∞(Ij),   j=1 j=1 [ X follows by a standard monotone-convergence  argument: Since

∞ k ∞ ν∞ Ij = lim ν Ij ,   k   j=1 →→∞ j=1 [ [     (2.10) implies that it suﬃ ces to show that

∞ ∞ k ν∞(Ij) = lim ν (Ij). (2.11) k j=1 →∞ j=1 X X

∞ k Because, for any k, the inﬁnite series ν (Ij) converges and, moreover, j=1 k X the terms ν (Ij) are non-increasing in k, it follows that, for each ε > 0, there c exists a ﬁnite set Jε, with complement Jε , such that

k ν (Ij) < ε j Jc X∈ ε for all k and, hence, by monotonicity,

ν∞(Ij) < ε. j Jc X∈ ε For ﬁxed ε, since Jε is ﬁnite,

k ν∞(Ij) = lim ν (Ij). k j Jε →∞ j Jε X∈ X∈ Hence

∞ ∞ k k ν∞(Ij) lim ν (Ij) ν∞(Ij) lim ν (Ij) 2ε. k k − ≤ c − c ≤ j=1 →∞ j=1 j Jε →∞ j Jε X X X∈ X∈

6 Since ε > 0 is arbitrary, (2.11) follows. u Because the cardinal of 0(X ) is c, the cardinal of I is at most c. By B Assumption 1, it follows that the atomless measure ν∞, which is deﬁned on ˆ ˆ ˆ 7 all subsets I I, must be trivial, i.e., that ν∞(I) = 0 for all I I. In ⊂ k ⊂ particular, ν∞(I) = 0. By (2.5) and (2.6), therefore, limk µ(UI ) = 0. By →∞ k ∞ k u the monotonicity of the sequence U , one also has U 0(X ) { I }k∞=1 I ∈ B k=1 and therefore \

∞ k ∞ k k µ∗ UI = µ UI = lim µ(UI ) = 0 (2.12) k k=1 ! k=1 ! ∞ \ \

To complete the proof, I note that, for any x UI = Bi, there ∈ i I k k [k∈ exists i I such that π (x) π (Bi) for all k, hence x U for all k . ∈ ∈ ∈ I ∞ k Thus UI U . By the monotonicity of outer measure, it follows that ⊂ I k=1 \ µ∗(UI ) = 0.

u u Proposition 2.4 For any measure µ 0(X ) and any family of ρ - ∈ M G open balls covering X, there exists a countable subfamily ∗ such that µ(G ) = 1, where G is the union of the sets in . G ⊂ G ∗ ∗ G∗ Proof. The proof follows along similar lines as the proof of Theorem 2, p. 235, in Billingsley (1968). Let µ and be as speciﬁed in the proposition. Because Xu is a metric space, TheoremG 4.21, p. 129, in Kelley (1955), implies that has a σ-discrete reﬁnement, i.e., there exists a family of ρu-open setsG (not necessarily balls) covering X such that, for every H H , there exists G(H) such that H G(H) and, moreover, can be written∈ H as a countable union∈ G ⊂ H ∞ = t H H t=1 [ where, for any t, any two sets Hti,Htj in t, there exists εt > 0 such that u i j i j H ρ (x , x ) εt for all x Hti and all x Htj. ≥ ∈ ∈ For any t, let It be the set of indices i such that Hti t. Moreover, ∈ H u let I be the set of indices i It such that µ(Bti) > 0 for some ρ -open t∗ ∈ 7 The underlying idea of this proof is due to Banach (1930), see also the proof of Theorem I in Marczewski and Sikorski (1948).

7 ball Bti Hti, and let be the family of sets Hti, i I . I claim that, for ⊂ Ht∗ ∈ t∗ any t, t∗ has at most countably many elements. To prove this claim, I note that H can be written as a countable union, Ht∗

∞ ∗ = ∗ , Ht Htn n=1 [ where for any n,

1 u ∗ = Hti ∗ µ(Bti) > for some ρ -open ball Bti Hti . Htn { ∈ Ht | n ⊂ } I claim that, for any n, the set has no more than n elements. For sup- Htn∗ pose that, contrary to the claim, tn∗ has n0 > n elements, Hti1 ,Hti2 , ..., Htin , H u 0 and let Bti1 ,Bti2 , ..., Btin be the associated ρ -open balls. Because the sets 0 u Hti1 ,Hti2 , ..., Hti are disjoint, so are the associated ρ -open balls Bti1 ,Bti2 , ..., Bti . n0 n0 The measure of the union of these balls is therefore equal to the sum n0 1 µ(Bti ) > n > 1. This conclusion is incompatible with the fact that j 0 · n j=1 X this measure cannot exceed µ(X) = 1. The assumption that tn∗ has n0 > n elements has thus led to a contradiction and must be false.H Hence has Htn∗ ∞ no more than n elements, and the countable union = also has Ht∗ Htn∗ n=1 no more than countably many elements. [ ∞ Now consider the union = . As a countable union of countable H∗ Ht∗ t=1 [ sets, ∗ is also countable. Recalling that, for each H , there exists G(H)H such that H G(H), deﬁne so that G∈ H if and only ∈ G ⊂ G∗ ⊂ G ∈ G∗ if G = G(H) for some H ∗. Since ∗ is countable, so is ∗. Moreover, ∈ H Hu G since the elements of belong to 0(X ), so does the countable union G∗ B

G∗ = G(H). H [∈H∗

I claim that µ(G∗) = 1. For suppose that µ(G∗) < 1. Then also µ(X G∗) > u \ 0. (Since G belongs to 0(X ), so does X G .) Hence also µ (X G ) > 0, ∗ B \ ∗ ∗ \ ∗ where µ∗ is the outer measure deﬁned by µ , in accordance with (2.1) above. Since H G(H) for all H, we also have X G X H , where H is the ⊂ \ ∗ ⊂ \ ∗ ∗ union of the sets Hti over all t and all i It∗. Thus, µ∗(X G∗) > 0 implies µ (X H ) > 0. Because the reﬁnement∈ of the family\ covers X , ∗ \ ∗ H G

8 ˆ the set X H∗ is a subset of the union H of the sets Hti over all t and all ˆ \ ˆ i It where, for any t, It := It It∗. By the monotonicity of outer measure, ∈ \ ˆ it follows that µ∗(X H∗) > 0 implies µ∗(H) > 0. \ˆ I claim that µ∗(H) > 0 is incompatible with Lemma 2.3. For this purpose, for any t, let Hˆt be the union of the sets Hti over all i Iˆt. Because u ∈ the sets Hti for i Iˆt are at ρ -distance εt from each other, it follows that u ∈ any ρ -open ball Bˆt that is contained in Hˆt is in fact contained in Hti for u some i Iˆt. By the deﬁnition of Iˆt, however, any ρ -open ball Bˆt that is ∈ contained in Hti for some i Iˆt.satisﬁes µ(Bˆti) = 0. By Lemma 2.3, it ˆ ∈ follows that µ∗(Ht) = 0 for all t. By the countable subadditivity of outer 8 ˆ measure, therefore, µ∗(H) > 0. The assumption that µ(G∗) < 1 has thus led to a contradiction and must be false.

u Proposition 2.2, i.e., the quasi-separability of the measures in 0(X ), follows immediately. M

3 Metrizability of the Topology of Weak Conver- u gence on 0(X ) M u Given the quasi-separability of measures in 0(X ), the proof that the u M topology of weak convergence on 0(X ) is metrizable is basically the same as the proof of the analogous resultM for Borel measures, e.g., Theorem 5, p. 238, in Billingsley (1968). Some modiﬁcations are needed to take account of u u the fact that 0(X ) is not a Borel σ-algebra and the measures in 0(X ) are quasi-separable,B rather than separable. M The following lemma is fundamental to the argument.

u Lemma 3.1 If a set B X belongs to 0(X ), the set ⊂ B ε u B := x0 X ρ (x0, x) < ε for some x B (3.1) { ∈ | ∈ } u also belongs to 0(X ). B u Proof. Consider the class 0(X ) of sets for which the lemma is true. It is easy to see that containsC ⊂ B the ﬁnite-dimensional cylinder sets. C Moreover, is closed under countable unions: If Br, r = 1, 2, ..., is any countable familyC of sets in , a point x belongs to the ε-neighbourhood of C 8 See Lemma 3.1.5, p. 89, in Dudley (2003).

9 ε rBr if and only if it belongs to Br for some r. The ε-neighbourhood of rBr ∪ ε ε ∪ u is therefore equal to the union rBr . Since Br implies Br 0(X ) u ∪ ∈ C ε ∈ B u and 0(X ) is closed under countable unions, it follows that rB 0(X ) B ∪ r ∈ B and hence that rBr . ∪ ∈ C Finally, is also closed under countable intersectons: If Br, r = 1, 2, ..., is any countableC family of sets in , a point x belongs to the ε-neighbourhood C ε of rBr if and only if it belongs to Br for all r. The ε-neighbourhood of ∩ ε rBr is therefore equal to the intersection rBr . Since Br implies ∩ ε u u ∩ ∈ C Br 0(X ) and 0(X ) is closed under countable intersections, it follows ∈ B ε Bu that rBr 0(X ) and hence that rBr . ∩ ∈ B u ∩ ∈ C Since 0(X ) is the smallest σ-algebra that is closed under countable unions andB countable intersections and that contains the ρu-open balls in u X, it follows that = 0(X ). C B u u Lemma 3.1 implies that, for any µ 0(X ) and B 0(X ), not only µ(B) but also µ(Bε) is well deﬁned∈ for M any ε > 0. One∈ can B therefore u deﬁne the Prohorov metric on 0(X ) by specifying the distance between M u any two measures µ and µˆ in 0(X ) as the greatest lower bound on the set of ε > 0 such that M

µ(B) µˆ(Bε) + ε and µˆ(B) µ(Bε) + ε (3.2) ≤ ≤ u for all sets B 0(X ). ∈ B Proposition 3.2 Under Assumption 1, the topology of weak convergence u u on 0(X ) and the topology on 0(X ) that is induced by the Prohorov metricM are equivalent. M

The proof of the analogous result in Billingsley (1968) comes in two distinct steps. The first step (Theorem 3, p. 236) specifies several families of sets of measures and shows that each family is a base for the topology of weak convergence. The second step uses this finding to prove the equivalence of the topology of weak convergence with the topology generated by the Prohorov metric. The argument here has the same structure. For the first step of the argument, I note that the family 0 of sets taking the form F

u ν 0(X ) fi(x)dν(x) fi(x)dµ(x) < ε, i = 1, ..., k (3.3) ∈ M | u − u ZX ZX

10 u u for some µ 0(X ), ε > 0, and f1, ..., fk in CB0(X ) is a base for the ∈ M u topology of weak convergence on 0(X ). I also consider families 1, 2, 3 of sets taking the forms M F F F

u ν 0(X ) ν(Fi) < µ(Fi) + ε, i = 1, ..., k (3.4) { ∈ M | } u u for some µ 0(X ), ε > 0, and ρ -closed sets F1, ..., Fk belonging to u ∈ M 0(X ) in the case of 1, B F u ν 0(X ) ν(Gi) < µ(Gi) + ε, i = 1, ..., k (3.5) { ∈ M | } u u for some µ 0(X ), ε > 0, and ρ -open sets G1, ..., Gk belonging to u ∈ M 0(X ) in the case of 2, and B F u ν 0(X ) ν(Ai) µ(Ai) < ε, i = 1, ..., k (3.6) { ∈ M | | − | } u for some µ 0(X ), ε > 0, and µ-continuity sets A1, ..., Ak belonging to u ∈ M 0(X ) in the case of 3. Each one of the families 1, 2, 3 is the base B F u F F F for a topology on 0(X ). The following result provides an analogue of Theorem 3, p. 236,M in Billingsley (1968).

Proposition 3.3 Each of the families 1, 2, 3 is a base for the topology u F F F of weak convergence on 0(X ). M Proof. The proof has two parts. The first part shows that the topologies induced by the families 1, 2, 3 are equivalent to each other. The second F F F part shows that the topology induced by the family 1 is equivalent to F the topology induced by the family 0. For the first part, I refer to the argument of Billingsley (1968, p. 237),F which goes through without any change. Because of Lemma 3.1, the requirement that the sets Fi,Gi,Ai u must all belong to 0(X ) plays no role in the argument. B For the second part, the requirement that the sets Fi in 1 must belong u F to 0(X ) does play a role. Therefore I give the adapted argument in detail. B I first show that any element of 1 contains an element of 0 as a subset. F u F u Any set N 1 takes the form (3.4) for some µ 0(X ), ε > 0, and ρ - ∈ F u ∈ M closed sets F1, ..., Fk in 0(X ). By Lemma 3.1, for any δ > 0, the sets δ u u B u Fi = x X ρ (x, Fi) < δ , i = 1, ..., k, also belong to 0(X ). Let δ > 0 { ∈ | } δ B be such that, for i = 1, ..., k, the set Fi is a µ-continuity set and, moreover, ε µ(F δ) < µ(F ) + . (3.7) i i 2

11 Next, let ϕ : R+ [0, 1] be a continuous function such that, for any → t R, ∈ ϕ(t) = 1 t if t [0, 1) and ϕ(t) = 1 if t 1. − ∈ ≥ u For any i, deﬁne a function fi : X [0, 1] by setting → 1 u fi(x) = ϕ ρ (x, Fi) , (3.8) δ · u u where again ρ (x, Fi) = minxˆ Fi ρ (x, xˆ). The function fi is obviously bounded u ∈ u and ρ -continuous. Since Fi 0(X ), By Lemma 3.1, fi is also measur- u 9∈ B u δ able with respect to 0(X ). We also have fi(x) = 0 for x X Fi and B u ∈ \ fi(x) = 1 for x Fi. For any ν 0(X ), therefore, ∈ ∈ M ε fi(x)dν(x) < fi(x)dµ(x) + u u 2 ZX ZX implies

ε δ ε ν(Fi) fi(x)dν(x) < fi(x)dµ(x) + µ(Fi ) + < µ(Fi) + ε ≤ u u 2 ≤ 2 ZX ZX u For any ν 0(X ) satisfying ∈ M ε fi(x)dν(x) fi(x)dµ(x) < u − u 2 ZX ZX for i = 1, ..., k, we therefore have

ν(Fi) < µ(Fi) + ε

u for i = 1, ..., k. Thus the set N 1 of measures ν 0(X ) that satisfy u ∈ F ∈ M (3.4) for the given µ 0(X ), ε > 0, and F1, ..., Fk contains a set measures u ∈ M u ν 0(X ) that satisfy (3.3) for the same µ 0(X ), ε > 0, and ∈ M u ∈ M µ 0(X ), ε > 0, and the speciﬁed functions fi, i = 1, ..., k. The latter ∈ M set an element of 0. Thus every element of 1 contains an element of 0. F F F For the claim that every element of 0 also contains an element of 1, the argument in Billingsley (1968) appliesF with hardly any change. ForF u example, let N 0 be the set of measures ν 0(X ) such that, for ∈u F u ∈ M given µ 0(X ), ε > 0, and f CB0(X ), ∈ M ∈ f(x)dν(x) f(x)dµ(x) < ε. (3.9) u − u ZX ZX 9 u u u u fi(x) > fi(ˆx) implies ρ (x, Fi) < ρ (ˆx, Fi) so that, for some η (ρ (x, Fi), ρ (ˆx, Fi)), x F η and xˆ Xu F η. ∈ ∈ i ∈ \ i

12 Without loss of generality, assume that f takes values in the unit interval. 1 u i Choose k0 so that k < ε and, for i = 1, ..., k0, let Fi = x X k f(x) . 0 u { u∈ | 0 ≤ } Then, for any i, Fi is ρ -closed. Moreover, since f is 0(X )-measurable, Fi u B is in 0(X ). By standard arguments, therefore, ν(Fi) < µ(Fi) + ε implies B 1 1 k0 1 1 k0 f(x)dν(x) < + ν(Fi) < + µ(Fi)+ε < f(x)dµ(x)+2ε. u k k k k u X 0 0 i=1 0 0 i=1 X Z X X Z By a parallel argument for the function x 1 f(x), there also exist u u → − ρ -closed, 0(X )-measurable sets Fi, i = k + 1, ..., 2k , such that, for i = B 0 0 k0 + 1, ..., 2k0, ν(Fi) < µ(Fi) + ε implies

(1 f(x))dν(x) < (1 f(x))dµ(x) + 2ε, u − u − ZX ZX or, equivalently,

f(x)dµ(x) < f(x)dν(x) + 2ε. u u ZX ZX Upon combining these arguments, one ﬁnds that the set N of measures u u ν 0(X ) that satisfy (3.9) for some given µ 0(X ), ε > 0, and ∈ M u ∈ M f CB0(X ), an element of 0, contains a subset consisting of measures ∈ F u that satisfy ν(Fi) < µ(Fi) + ε for the given µ, ε, k = 2k0, and ρ -closed, u 0(X )-measurable sets Fi, i = 1, ..., k. Thus N has a subset that belongs B to 1. FBy taking intersections of sets like N, with diﬀerent functions f, the argument can be generalized to all sets in 0. Thus every element of 0 F F contains an element of 1 as a subset. This completes the proof of the claim F that the topologies induced by 0 and 1 are equivalent. F F

u Proposition 3.4 For any measure µ 0(X ) that is quasi-separable, the topology induced by the Prohorov metric∈ and M the topology of weak convergence u on 0(X ) are equivalent at µ. M

Proof. Let N(µ) be any 1-neighbourhood of µ, characterized by ε > 0 u u F and ρ -closed, 0(X )-measurable sets Fi, i = 1, ..., k, such that ν N(µ) if B ∈ and only if ν satisﬁes (3.4) for µ, ε, and F1, ..., Fk. Let δ < ε be such that, for δ u u i = 1, ..., k, the set Fi = x X ρ (x, Fi) < δ is a µ-continuity set and, δ { ε∈ | } u moreover, µ(Fi ) < µ(Fi) + 2 . For any ν 0(X ) such that p(ν, µ) < δ, one has ∈ M δ ν(Fi) < µ(Fi ) + δ < µ(Fi) + ε,

13 so the p-open ball with radius δ around µ is a subset of N(µ). Therefore u the topology on 0(X ) that is induced by the Prohorov metric is at least as large at µ as theM topology of weak convergence. Next, suppose that µ is quasi-separable. I will show that for any ε > 0, the p-open ball with radius ε around µ contains an 3-neighbourhood N(µ) of µ. Fix a cover of Xu by ρu-open, µ-continuity ballsF with diameters less ε than δ, where δ < 3 . Appealing to quasi-separability, pass to a countable subfamily Bi such that µ( iBi) = 1. Construct disjoint µ-continuity { }i∞=1 ∪ stes A1,A2, ... by setting A1 = B1 and, for i > 1,Ai = Bi j 1 δ (3.10) ∪i=1 − and let be the set of unions of the sets Ai over subsets of the indices i = 1, ...,A k. Then each A is a µ-continuity set, and, by Proposition 3.3, ∈ A there is a neighbourhood N(µ) 3 of µ such that, for any ν N(µ), ∈ F ∈ ν(A) µ(A) < δ for all A . (3.11) | − | ∈ A Iclaim that N(µ) is contained in the p-open ball with radius ε around µ. u To prove this claim, consider any B 0(X ). Let IB be the set of indices i ∈ B u k such that Ai B = and let AB := i IB Ai. Then B AB (X i=1 Ai). ∩ 6 ∅ ∪ ∈ ⊂ ∪ \∪ Moreover, AB and, because the sets Ai all have diameters less than δ, δ ∈ A AB B . Using (3.10) and (3.11), one obtains ⊂ k δ ε µ(B) µ(AB) + 1 µ( Ai) < ν(AB) + 2δ ν(B ) + 2δ < ν(B ) + ε. ≤ − ∪i=1 ≤ Similiarly, taking account of the fact that (3.10) and (3.11) imply

k ν( Ai) > 1 2δ, ∪i=1 − one also obtains

k δ ε ν(B) ν(AB) + 1 ν( Ai) < µ(AB) + 3δ µ(B ) + 3δ < µ(B ) + ε. ≤ − ∪i=1 ≤ Thus, for any ν N(µ), the Prohorov distance between ν and µ is less than ∈ ε. The speciﬁed set N(µ) 3 is contained in the p-open ball with radius ∈ F u ε around µ. At µ, therefore the topology of weak convergence on 0(X ) is at least as large as the topology that is induced by the ProhorovM metric. Given that the topology induced by the Prohorov metric is also at least as large at µ as the topology of weak convergence, it follows that the two topologies are equivalent at µ.

Now Proposition 3.2 is a straightforward corollary to Propositions 2.2 and 3.4.

14 u u 4 Why Bother about 0(X ) and 0(X )? B M u Dudley’s initiative in introducing the σ-algebra 0(X ) and the space of u B measures 0(X ) was not much followed. For the purpose that he was interestedM in, the σ-algebra induced by the Skorokhod topology and the associated space of Borel measures served the same purpose. Moreover, u whereas the uniform topology, as well as 0(X ) and the space of measures u B 0(X ), are conceptually simpler, they suffer from the inconveniences of havingM to deal with non-Borel measures. Recently, however, the issue that the Borel σ-algebra (Xu) that is induced by the ρu-topology on X is “too large” has comeB up in another context. Recent developments in the theory of strategic games of incomplete information suggest that, for the product spaces under consideration there, u u a reliance on the σ-algebra 0(X ) and the space of measures 0(X ) may be necessary. B M Incompleteness of information in a strategic game concerns not only an agent’s information (probabilistic beliefs) about the exogenous parameters of the game, but also the agent’sinformation about the other agents’information about the exogenous parameters of the game, the agent’sinformation about the other agents’infomation about everybody else’information about the exogenous parameters of the game, and so on, in a seemingly intractable infinite regress. To avoid having to deal with the seemingly intractable hierarchy of different orders of probabilistic beliefs about exogenous parameters and other agents’beliefs of different orders, Harsanyi (1967/68) introduced the notion of an agent’s abstract type, together with a belief function that maps the agent’s abstract types into probability measures over exogenous parameters and other agents’abstract types. This formalism provides a workable apparatus for the analysis of strategic interdependence with incomplete information in many areas of applied work. However, it suffers from the fact that the notion of an abstract type space is a black box and the notion of beliefs as type-determined probability measures over exogenous data and other agents’abstract types involves an element of circularity. These shortcomings were remedied by Mertens and Zamir (1985), Bran- denburger and Dekel (1993), and Heifetz (1993). These authors showed that one can actually interpret any hierarchy of beliefs of different orders of an agent as a “type”in the Harsanyi sense. Moreover, Kolmogorov’sextension theorem implies that any belief hierarchy in which the beliefs of different orders are mutually consistent determines a unique measure on the product σ-algebra for space of exogenous parameters and of other agents’belief hi-

15 erarchies. The infinite regress in belief formation is shown to be harmless when the model is formulated in terms of infinite-dimensional products of spaces of beliefs of different orders. If the spaces of consistent belief hierarchies are given the product topology and if the space of probability measures on exogenous parameters and other agents’belief hierarchies is given the associated topology of weak convergence, the mapping from the space of an agent’s belief hierarchies to probability measures over exogenous parameters and other agents’ belief hierarchies that is given by Kolmogorov’s theorem is a homeomorphism. Moreover, for any abstract type space model à la Harsanyi, if belief functions are continuous, there is a natural embedding of the different agents’ abstract type spaces in the corresponding spaces of belief hierarchies with the product topology. The product topology on the space of belief hierarchies is well suited to capturing upper hemi-continuity properties of optimal choices, e.g., best responses to other agents’strategies; it is also well suited to capturing upper hemi-continuity properties of outcomes under various equilibrium concepts (Bayes-Nash or rationalizable). But it is not well suited to capturing lower hemi-continuity properties of strictly ε-rationalizable outcomes. The rea- son is that the product topology gives decreasing weight to beliefs of very high order in the hierarchy of beliefs, but in some games, optimizing or ε-optimizing behaviour can be sensitive to changes in beliefs of arbitrarily high orders in the hierarchy.10 This shortcoming of the product topology has induced Dekel et al. (2006) and Chen et al. (2010, 2017) to propose finer topologies for the spaces of belief hierarchies. Chen et al. treat the space of an agent’s consistent belief hierarchies as a subspace of the product of the spaces of beliefs of different orders and impose a uniform metric on this space. Dekel et al. (2006) specify finer topologies directly in terms of the desired continuity properties of strategic behaviour, but Chen et al. (2010, 2017) show that these topologies can be reinterpreted in terms of their own uniform weak topology. What then becomes of the Mertens-Zamir reconciliation of an approach based on belief hierarchies with the Harsanyi abstract-type-space approach? In particular, what becomes of the proposition that Kolmogorov’s extension theorem can be understood as defining a homeomorphism between the space of belief hierarchies and the space of probability measures on exgenous parameters and other agents’belief hierarchies?

10 See Dekel et al. (2006, 2007), Chen et al. (2010, 2017).

16 This is not just a question about the impact of having a finer topology for the domain of the mapping. It is also a question about the impact of having a finer topology on the range; indeed, it is even a question about the specification of the range itself. Kolmogorov’s extension theorem gives measures on the product σ-algebra for the space of exogenous parameters and other agents’belief hierarchies or, equivalently measures on the Borel σ-algebra for the product topology. The Borel σ-algebra for the uniform weak topology is strictly larger. How then does one deal with the discrepancy between the domain of the measures defined by Kolmogorov’stheorem? One approach might be to try and extend the measures to the larger σ-algebra. Because the infinite- dimensional product space with a uniform metric is non-separable, the first approach does not work. A measure on the product σ-algebra can always be extended to a measure on the Borel σ-algebra for the finer topology but the extension need not be unique and it need not be countably additive.11 An alternative approach restricts the measures in question to a σ-algebra u on X that is coextensive with the product σ-algebra. The σ-algebra 0(X ) that is generated by the ρu-open spheres in X serves this purpose veryB well because it is in fact coextensive with the product σ-algebra (X). The u u u B space 0(X ) of probability measures on (X , 0(X )) is coextensive with the spaceM (X) of probability measures on (X,B (X)). To make use of this observationM in studying the relation between the spaceB of belief hierarchies of an agent and the space of probability measures over exogenous parameters and other agents’belief hierarchies, one needs to understand the topology u of weak convergence on 0(X ). M 11 For the existence of a finitely additive extension, see Bachman and Sultan (1980). The suffi cient conditions they give for uniqueness and countable additivity are not satisfied here. To see the problem, set Xk = 0, 1 in (1), and let µ (X) be the product 1 { } ∈ M measure that assigns probability 2 to each of the events xk = 0 and xk = 1 , for all k. By Theorem 2.1, p. 290, of Bachman and Sultan (1980),{ this measure} { can be extended} to a finitely additive measure on the algebra generated by the lattice of ρu-closed subsets of X. However, the values of the extension for the set A0 of sequences x X such that ∈ xk = 1 no more than finitely many times and for the set A1 of sequences x X such ∈ that xk = 0 no more than finitely many times are to some extent arbitrary because, in the product topology on X, the empty set is the largest closed set that is contained in A0 or A1 and X itself is the smallest closed set that contains A0 or A1 . Thus the extended measure µ∗ might have µ∗(A0) = 0 and µ∗(A1) = 1 or µ∗(A0) = 1 and µ∗(A1) = 0. u As for the possibility that µ∗ might be countably additive, notice that the ρ -topology on 0, 1 ∞ is equivalent to the discrete topology, so the associated Borel σ-algebra is { } the power set. Moreover, µ∗ assigns probability zero to the singletons x 0, 1 ∞. { } ⊂ { } Countable additivity of µ∗ would be incompatible with the axiom that the cardinal c of the continuum is not atomlessly measurable.

17 This is where Propositions 2.2 and 3.2 are useful. In Hellwig (2016/2020), I use these propositions to show that, with the uniform weak topology on the spaces of belief hierarchies, Kolmogorov’s extension theorem still provides a homeomorphism between the space of an agent’s belief hierarchies and the space of probability measures over exogenous parameters and other agents’belief hierarchies (with the topology of weak convergence). Without going into the game theoretic details, the argument boils down down to the following. For ` = 1, 2, ..., let π` be the projection from X to the ﬁnite product

` ` X = Xk. (4.1) k=1 Y u For any µ 0(X ) and any `, let ∈ M ` ` 1 ` Π (µ) = µ (π )− X , (4.2) ◦ ∈ M and let 1 2 Π∞(µ) = (Π (µ), Π (µ), ...). (4.3)

Proposition 4.1 Suppose that each of the spaces X` , ` = 1, 2, ...,is M ∞ ` endowed with the Prohorov metric p`, and that the product space X M `=1 is endowed with the uniform metric du such that, for any twoY sequences µ` , µˆ` , { }`∞=1 { }`∞=1 u ` ` ` ` d ( µ `∞=1, µˆ `∞=1) = sup d`(µ , µˆ ). (4.4) { } { } `

u If 0(X ) has the topology of weak convergence, then, under Assumption M u 1, the mapping Π is a homeomorphism between 0(X ) and the subspace ∞ M ∞ Hu X` that consists of those sequences µ` that are mutually ⊂ M { }`∞=1 `=1 consistentY in that µ`+1 (π`) 1 = µ` for all `. ◦ −

u u ∞ ` Proof. The mapping Π from 0(X ) to H X is obviously ∞ M ⊂ M `=1 Y 1 injective and onto. By Kolmogorov’sextension theorem, the inverse (Π∞)− u u from H to 0(X ) is also injective and onto. Therefore it suﬃ ces to show M 1 that Π∞ and (Π∞)− are both continuous.

18 ` ` Continuity of Π∞ is straightforward: For any ` and any set W (X ), the cylinder set ∈ B ` ` Wˆ = W X`+1 X`+2 ... × × × u belongs to 0(X ), and, for any ε > 0, the cylinder set B `ε ` ε Wˆ = (W ) X`+1 X`+2 ... × × × that is deﬁned by the ε-neighbourhood (W `)ε of W ` in X` is an ε-neighbourhood ` u u of Wˆ in X . If the Prohorov distance between two measures µ, µˆ in 0(X ) is less than ε, we have M µ(Wˆ `) < µˆ(Wˆ `ε) + ε and µˆ(Wˆ `) < µ(Wˆ `ε) + ε. By the deﬁnition of the marginal distributions µ` = Π`(µ) and µˆ` = Π`(ˆµ), it follows that µ`(W `) < µˆ`(W `ε) + ε and µˆ`(W `) < µ`(W `ε) + ε. Because the choice of W ` (X`) was arbitrary, it follows that the Pro- ` ` ∈ B ` ` ` ` ` u horov distance d`(µ , µˆ ) between µ = Π (µ) and µˆ = Π (ˆµ) in Π ( 0(X )) is no greater than ε. Since ε may be taken to be arbitrarily close toM the Pro- u horov distance p(µ, µˆ) between µ and µˆ in 0(X ), it follows that the ` ` M` ` ` u Prohorov distance between µ = Π (µ) and µˆ = Π (ˆµ) in Π ( 0(X )) is no greater than p(µ, µˆ). Since this is true for all `, it follows thatM

u ` ` d (Π∞(µ), Π∞(ˆµ)) = sup d`(Π (µ), Π (ˆµ)) p(µ, µˆ). ` ≤ u u Continuity of the map Π∞ from 0(X ) to H follows immediately. M 1 u u Next, consider the map β := (Π∞)− from H to 0(X ) that is given by Kolmogorov’s extension theorem. Proceeding indirectly,M suppose that β r `r is not continuous. Then there exist sequences h = µ `∞=1, r = 1, ..., , ` u r { }u r ∞ and h = µ `∞=1 in H such that h converges to h H but β(h ) does { } u r ∈ not converge to β(h) in 0(X ). Convergence of h to h implies that M `r ` lim sup d`(µ , µˆ ) = 0. (4.5) r →∞ ` Non-convergence of β(hr) to β(h) implies that, for some ε > 0 and some subsequence hr0 of hr , { } { } p(β(hr), β(h)) ε ≥ 19 u for all r . Thus, for every r , there exists a set Br 0(X ) such that 0 0 0 ∈ B r ε β(Br h ) > β(B h) + ε (4.6) 0 | r0 | or ε r β(Br h) > β(B h ) + ε, (4.7) 0 | r0 | ε u u where Br 0(X ) is the ε-neighbourhood of Br0 in X . 0 ∈ B ` ` ` ` u For any `, let B = π (Br ) be the projection of Br to X = π (X ) r0 0 0 and let (B` )ε be an ε-neighbourhood of B` in X`. Let r0 r0 ` ` Bˆ := B X`+1 X`+2 ... r0 r0 × × × and ` ε ` ε (Bˆ ) := (B ) X`+1 X`+2 ... r0 r0 × × × be the cylinder sets in Xu that are deﬁned by B` and (B` )ε. One easily r0 r0 veriﬁes that the sequences Bˆ` and (Bˆ` )ε are nonincreasing and that { r0 } { r0 } ∞ ∞ B = Bˆ` and Bε = (Bˆ` )ε (4.8) r0 r0 r0 r0 `=1 `=1 \ \ 12 for all r0. By elementary measure theory, for any r0 and any δ > 0, there exists an integer Lr0 (δ) such that, for ` > Lr0 (δ),

β(Bε h) β((Bˆ` )ε h) δ (4.9) r0 | ≥ r0 | − and β(Bε hr0 ) β((Bˆ` )ε hr0 ) δ. (4.10) r0 | ≥ r0 | − Moreover, by (4.8), ` β(Br h) β(Bˆ h) (4.11) 0 | ≤ r0 | and r0 ` r0 β(Br h ) β(Bˆ h ). (4.12) 0 | ≤ r0 | ε Set δ = 2 and combine (4.9) - (4.12) with (4.6) and (4.7). Thereby one r ﬁnds that, for all r0, all δ > 0, and all ` > L 0 (δ), either ε β(Bˆ` hr0 ) > β((Bˆ` )ε h) + (4.13) r0 | r0 | 2 or ε β(Bˆ` h) > β((Bˆ` )ε hr0 ) + . (4.14) r0 | r0 | 2 12 Theorem 3.1.1, p. 86, in Dudley (2002).

20 ε ˆ` ˆ` ε r0 Since (B ) 2 (B ) , it follows that, for all r0, all δ > 0, and all ` > L (δ), r0 r0 either ⊂ ` r ` ε ε β(Bˆ h 0 ) > β((Bˆ ) 2 h) + (4.15) r0 | r0 | 2 or ` ` ε r ε β(Bˆ h) > β((Bˆ ) 2 h 0 ) + . (4.16) r0 | r0 | 2 1 2 By the deﬁnition of β as the inverse of Π∞ = (Π , Π , ...), and the ` ε cylinder nature of the sets Bˆr and (Bˆ ) 2 , we also have 0 r0

β(Bˆ` hr0 ) = Π`(B` β(hr0 )) = µ`r0 (B` ), (4.17) r0 | r0 | r0 β(Bˆ` h) = Π`(B` β(h)) = µ`(B` ), (4.18) r0 | r0 | r0 ` ε r ` ` ε r `r ` ε β((Bˆ ) 2 h 0 ) = Π ((B ) 2 β(h 0 )) = µ 0 ((B ) 2 ), (4.19) r0 | r0 | r0 ` ε ` ` ε ` ` ε β((Bˆ ) 2 h) = Π ((B ) 2 β(h)) = µ ((B ) 2 ). (4.20) r0 | r0 | r0 Thus, (4.15) and ??) can be rewritten as

`r ` ` ` ε ε µ 0 (B ) > µ ((B ) 2 ) + (4.21) r0 r0 2 and ` ` `r ` ε ε µ (B ) > µ 0 ((B ) 2 ) + , (4.22) r0 r0 2

ε r0 and one of (4.21), (4.22) must hold if r0, δ = 2 , and ` > L (δ). But then, `r ` ε for such r , δ, and `, d`(µ 0 , µˆ ) , contrary to (4.5). The assumption that 0 ≥ 2 β = Π∞ is not continuous has thus led to a contradiction and must be false.

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