Measurable selection theorems and probabilistic control models in general topological spaces

I. V. Evstigneev

Sbornik. Mathematics, Tome 59 (1988) no. 1, pp. 25-37

Résumé

Let $(\Omega,\mathscr F)$ be a measurable space, $P$ a finite measure on $\mathscr F$, and $X$ a $\sigma$-compact topological space (not necessarily metrizable); $\mathscr B(X)$ is the Baire $\sigma$-algebra of $X$ and $\mathbf B(X)$ the Borel $\sigma$-algebra. Let $\mathscr F^P$ be the completion of $\mathscr F$ with respect to the measure $P$ and $\sigma(\mathscr A(\mathscr F))$ the $\sigma$-algebra generated by the sets $\Delta\subseteq\Omega$ representable in the form $\Delta=\mathrm{pr}_\Omega D$, where $D\subseteq\Omega\times[0,1]$ and $D\in\mathscr F\times\mathbf B([0,1])$. A mapping $\xi\colon\Delta\to X$ is called a selection of a set $\Gamma$ if $(\omega,\xi(\omega))\in\Gamma$ for $\omega\in\mathrm{pr}_\Omega\Gamma$. The central result (a measurable selection theorem) is the following. Theorem 1. For any set $\Gamma\in\mathscr F\times\mathscr B(X)$ there exist measurable mappings $$ \xi\colon(\Omega,\mathscr F^P)\to(X,\mathbf B(X)),\qquad\eta\colon(\Omega,\sigma(\mathscr A(\mathscr F)))\to(X,\mathscr B(X)), $$ which are selections for $\Gamma$. The proof of the existence of $\eta$ is based on the continuum hypothesis. Theorem 1 (the part concerning the existence of $\xi$) is used to obtain necessary and sufficient conditions for an extremum in certain problems involving control of random processes with discrete time. Bibliography: 34 titles.