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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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


Authors: D. Landers and L. Rogge
Journal: Proc. Amer. Math. Soc. 79 (1980), 471-473
MSC: Primary 60F15
MathSciNet review: 567995
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Abstract: Let $ {L_s}(\mu )$ be the space of real valued random variables with $ \mu (\vert f{\vert^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\vert C)$ with $ {\left\Vert {f - {\mu _s}(f\vert C)} \right\Vert _s} \leqslant {\left\Vert {f - c} \right\Vert _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\vert{C_n}) \to {\mu _s}(f\vert{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.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1980-0567995-4
PII: S 0002-9939(1980)0567995-4
Keywords: Pointwise convergence, projections on closed convex sets, conditional expectations
Article copyright: © Copyright 1980 American Mathematical Society