Consistent Families of Measures and Their Extensions

Let $\Sigma $ be a family of Borel fields of subsets of a set S and $\mu_\mathfrak{S} $ probabilistic measures on measurable spaces $\langle {\mathfrak{S},S} \rangle $, where $\mathfrak{S} \in \Sigma $. The family of measures $\mu_\mathfrak{S} $, $\mathfrak{S} \in \Sigma $ is denoted by $\mu_\Sigma $.

The measures $\mu_{\mathfrak{S}_1 } $ and $\mu_{\mathfrak{S}_2 } $ are said to be consistent if $\mu_{\mathfrak{S}_1 } (A) = \mu_{\mathfrak{S}_2 } (A)$ for any $A \in \mathfrak{S}_1 \cap \mathfrak{S}_2 $. If any pair of measures of the family $\mu_\Sigma $ is consistent, the family itself is referred to as consistent.

The consistent family $\mu_\Sigma $ is said to be extendable if there is a measure $\mu_{[\Sigma ]} $ on the measurable space $\langle {[\Sigma ],S} \rangle $ consistent with each measure of $\mu_\Sigma $ ($[\Sigma ]$ is the smallest Borel field containing all $\mathfrak{S} \in \Sigma $).

For the purposes of the theory of games the following special case of extendability is important. Let ${\bf \mathfrak{K}}$ be a finite complete complex and M the set of its vertices. Let a finite set $S_a $ correspond to each vertex a of ${\bf \mathfrak{K}}$ and the set $S_A = \Pi _{\alpha \in A} S_\alpha $ to each subset $A \subset M$. Let \[ \mathfrak{S}_K = \left\{ {X_K :X_K = Y_K \times S_{M - K},\, Y_K \subset S_K } \right\},\quad K \in {\bf \mathfrak{K}};\]$\mu _K $ is a measure on $\left\langle {\mathfrak{S}_K,S_M } \right\rangle $ and $\mu _{\bf \mathfrak{K}} $ is the family of all such measures. The extendability of the family $\mu _{\bf \mathfrak{K}} $ is closely related with the combinatorial properties of the complex ${\bf \mathfrak{K}} $.

Any maximal face of the complex ${\bf \mathfrak{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 ${\bf \mathfrak{K}} $). If T is an extreme face of ${\bf \mathfrak{K}} $ the complex ${\bf \mathfrak{K}}^* $ obtained by removing from ${\bf \mathfrak{K}} $ all proper vertices of T with their stars is said to be a normal subcomplex of ${\bf \mathfrak{K}} $. A complex ${\bf \mathfrak{K}} $ is said to be regular if there is a sequence \[ {\bf \mathfrak{K}} = {\bf \mathfrak{K}}_0 \supset {\bf \mathfrak{K}}_1 \supset \cdots \supset {\bf \mathfrak{K}}_n\] of subcomplexes of ${\bf \mathfrak{K}} $ where ${\bf \mathfrak{K}}_i $ is a normal subcomplex of ${\bf \mathfrak{K}}_{i - 1},i = 1, \cdots,n$, and the last member vanishes.

The main results of the paper consists in the following statement.

Theorem. The regularity of the complex${\bf \mathfrak{K}}$is a necessary and sufficient condition of extendability of any consistent family of$\mu_{\bf \mathfrak{K}}$of measures.