Discussion Papers of the Max Planck Institute for Research on Collective Goods 2017/6 MAX PLANCK SOCIETY Probability Measures on Product Spaces with Uniform Metrics 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 -algebra generated by the open balls, which was introduced by Dudley (1966), coincides with the product -algebra. Any probability measure on the product space with this -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 conver- gence of non-Borel measures, -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 met- rics 1, 2, ... Suppose that the product 1 X = Xk (1.1) k=1 Y has the topology induced by the uniform metric u where, for any x and x^ in X, u (x, x^) = sup k(k(x), k(^x)) (1.2) k u and k is the projection from X to Xk. I use the notation X to indicate u u that X has the topology induced by . Let 0(X ) be the -algebra that is u B u generated by the -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, - B u continuous, and 0(X )-measurable real-valued functions on X. Say that r B u a sequence of measures in 0(X ) converges weakly to a measure uf g M 0(X ) if and only if 2 M f(x)dr(x) f(x)d(x) (1.3) u ! u ZX ZX u for all f CB0(X ). 2 u u This paper shows that, even though X is non-separable and 0(X ) is not a Borel -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 theM property that any covering of the set by open balls contains a count- able 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 -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 -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 -algebra induced by the open balls, he avoided the diffi culties arising from the large size of the Borel -algebra. Dudley’s approach pro- vides an alternative to the use of the Skorokhod topology and associated Borel -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 conver- gence 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 -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 -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 be- cause 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 car- dinal of X is at least 2! = c where ! 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 -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 -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 ! or c.5 u The cardinal of the -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 X^ = x = (x1, x2,...) X xk x , x for all k . Clearly, X^ X has the f 2 j 2 f k kg g cardinal c, and so does any subset of X^, or of X, that is u-dense. Throughout the analysis, I impose the following assumption. Assumption 1 The cardinal c of the continuum is not atomlessly mea- surable: 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 0(X ) is called quasi-separable if, for any family of u-open balls that2 covers M X, there exists a countable subfamily suchG that the union G of the sets in satisfies (G ) = 1. G G u Proposition 2.2 Under Assumption 1, any measure 0(X ) is quasi- separable. 2 M The proof of Proposition 2.2 makes use of the outer measure that a u measure 0(X ) induces on every subset of X. For a given set U X, 2 M 6 the value (U) of the outer measure that is induced by is defined as 1 1 u (U) = inf (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 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 -spheres such that (Bi) = 0 for all i I. Thenf g the2 set 2 UI := Bi (2.2) i I [2 has the outer measure (UI ) = 0. 5 Theorem 13.1.1 in Dudley (2002). 6 See Dudley (2002), p. 89. 4 Proof.