Consistent Families of Measures and Their Extensions

Let be a family of Borel fields of subsets of a set S and S probabilistic measures on measurable spaces S, S, where S. The family of measures S, S is denoted by _. The measures ₒ䃑 and ₒ䃒 are said to be consistent if ₒ䃑 (A) = ₒ䃒 (A) for any A S₁ S₂. If any pair of measures of the family _ is consistent, the family itself is referred to as consistent. The consistent family _ is said to be extendable if there is a measure on the measurable space, S consistent with each measure of _ (is the smallest Borel field containing all S). 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 = ₀ S_ to each subset A M. Let \ SK = \ {XK: XK = YK S₌ - ₊, \, YK SK \}, K K;\ K is a measure on SK, SM and ₊ is the family of all such measures. The extendability of the family ₊ 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 = K₀ K₁ Kₙ\ of subcomplexes of K where Kᵢ is a normal subcomplex of K₈ - 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 complex Kis a necessary and sufficient condition of extendability of any consistent family of ₊of measures.