On Relations Between Strict-Sense and Wide-Sense Conditional Expectations

In this paper we investigate the relations between conditional expectations in the strict sense (i.e. ordinary conditional expectations) and in the wide sense (i.e. projections of a Hilbert space $L_2 $ into some closed linear subspace), as they were defined by Doob [3].Let $(X,F,\mu )$ be a probability space and $G \subset F$ be some $\sigma $-algebra. We denote by $L_2 (G)$ the system of all G-measurable random variables of $L_2 $. Following Bahadur we name such subspaces, $L_2 (G)$, measurable subspaces, and projections on them measurable projections.The most important result of the paper is the followingTheorem 3.The system$\mathfrak{M} \subset L_2 $is a measurable subspace if and only i f it satisfies the following conditions: (0) $\mathfrak{M}$ is a closed linear subspace, (1) $1 \in \mathfrak{M}$, (2) if$f \in \mathfrak{M}$, $g \in \mathfrak{M}$, then$\max (f,g) \in \mathfrak{M}$.In papers published up to now the operation of multiplying two functions was used, but it was introduced only for bounde...