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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.

References [Enhancements On Off] (What's this?)

  • [1] T. Ando and I. Amemiya, Almost everywhere convergence of prediction sequence in $ {L_p}(1 < p < \infty )$, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 4 (1965), 113-120. MR 0189077 (32:6504)
  • [2] R. E. Barlow, D. J. Bartholomew, J. M. Bremner and H. D. Brunk, Statistical inference under order restrictions, Wiley, New York, 1972. MR 0326887 (48:5229)

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Keywords: Pointwise convergence, projections on closed convex sets, conditional expectations
Article copyright: © Copyright 1980 American Mathematical Society

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