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 \m...