Extensions of measures and the von Neumann selection theorem

Abstract:

Let $(X,{B_X})$ be a Blackwell space, where ${B_X}$ is the $\sigma$-algebra of Borel sets. Then if $\sigma$ is a finite measure defined on a countably generated sub-$\sigma$-algebra $B \subset {B_X},\sigma$ can be extended to a Borel measure $\tau$. Equivalently, if $X$ and $Y$ are Blackwell and $f:X \to Y$ is Borel, and $\mu$ is a Borel measure carried on $f(X) \subset Y$, then there exists a Borel measure $\tau$ on $X$ with ${\tau ^f} = \sigma$, where ${\tau ^f}(E) = \tau ({f^{ - 1}}(E))$. We characterize $\{ \tau |{\tau ^f} = \sigma \}$ if $f$ is semischlicht.