A short proof for a.e. convergence of generalized conditional expectations

Abstract:

Let ${L_s}(\mu )$ be the space of real valued random variables with $\mu (|f{|^s}) < \infty ,1 < s < \infty$ . Let $C \subset {L_s}(\mu )$ be a closed convex set. For each $f \in {L_s}(\mu )$ there exists a unique element ${\mu _s}(f|C)$ with ${\left \| {f - {\mu _s}(f|C)} \right \|_s} \leqslant {\left \| {f - c} \right \|_s}$ for every $c \in C$. Let ${C_n}$ be a decreasing or increasing sequence of closed convex lattices converging to the closed convex lattice ${C_\infty }$. We show that ${\mu _s}(f|{C_n}) \to {\mu _s}(f|{C_\infty })\mu$-a.e. for every $f \in {L_s}(\mu )$. This result contains the results of a.e. convergence of prediction sequences of Ando-Amemiya and the result of Brunk and Johansen of a.e. convergence of conditional expectations given $\sigma$-lattices.