Regular conditional expectations of correspondences

E. B. Dynkin ; I. V. Evstigneev

Teoriâ veroâtnostej i ee primeneniâ, Tome 21 (1976) no. 2, pp. 334-347

Résumé

Let $f(\omega,z)$ be random variables depending on a parameter $z$. We construct a function $g(\omega,z)$ with the following property: $g(\omega,\varphi(\omega))$ is the conditional expectation with respect to $\mathscr G$, of $f(\omega,\varphi(\omega))$ for each $\mathscr G$-measurable $\varphi$. All the functions $g$ with the above property are indistinguishable (i. e. coincide for all $z$ almost surely). We call them regular conditional expectations of $f$. We also study functions with values in the set of all closed non-empty subsets of a finite-dimensional vector space. Basic theorems about regular conditional expectations are extended for all such set-valued functions (we call them correspondences). Until recently conditional expectations were considered only for convex-valued correspondences (see e. g. [3]). To eliminate this restriction, we use the notion of $\mathscr G$-atoms introduced by V. A. Rokhlin [13] (see also [11]), consider a decomposition of the space into disjoint $\mathscr G$-atoms and a non-atomic part and prove for the latter a generalization of well-known theorem of Lyapunov.