Consistent Families of Measures and Their Extensions

Let Σ be a family of Borel fields of subsets of a set S and μS probabilistic measures on measurable spaces S,S, where S ∈ Σ. The family of measures μS, S ∈ Σ is denoted by μ_Σ. The measures μS₁ and μS₂ are said to be consistent if μS₁ (A) = μS₂ (A) for any A ∈ S₁ ∩ S₂. If any pair of measures of the family μ_Σ is consistent, the family itself is referred to as consistent. The consistent family μ_Σ is said to be extendable if there is a measure μ[Σ ] on the measurable space [Σ ],S consistent with each measure of μ_Σ ([Σ ] is the smallest Borel field containing all S ∈ Σ). For the purposes of the theory of games the following special case of extendability is important. Let K be a finite complete complex and M the set of its vertices. Let a finite set Sₐ correspond to each vertex a of K and the set SA = Π α ∈ A S_α to each subset A ⊂ M. Let \[ S_K = \{ {X_K :X_K = Y_K × SM - K ,\, Y_K ⊂ S_K } \}, K ∈ { K};\]μ K is a measure on SK ,SM and μ K is the family of all such measures. The extendability of the family μ K is closely related with the combinatorial properties of the complex K. Any maximal face of the complex K is said to be an extreme face if it has proper vertices (i.e. such vertices which do not belong to any other maximal face of K). If T is an extreme face of K the complex K^* obtained by removing from K all proper vertices of T with their stars is said to be a normal subcomplex of K. A complex K is said to be regular if there is a sequence \[ { K} = { K}_0 ⊃ { K}_1 ⊃ ⋯ ⊃ { K}_n\] of subcomplexes of K where Kᵢ is a normal subcomplex of Ki - 1 ,i = 1, ⋯ ,n, and the last member vanishes. The main results of the paper consists in the following statement. Theorem. The regularity of the complexKis a necessary and sufficient condition of extendability of any consistent family ofμKof measures.