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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Characterization of conditional expectation operators for Banach-valued functions
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by D. Landers and L. Rogge PDF
Proc. Amer. Math. Soc. 81 (1981), 107-110 Request permission

Abstract:

Let $B$ be a Banach space. It is well known that for every probability space $(\Omega , \mathcal {A}, P)$ and every sub-$\sigma$-field $\mathcal {A}_0 \subset \mathcal {A}$ there exists a conditional expectation operator $P^{\mathcal {A}_0}X$ for $B$-valued $P$-integrable functions. This operator maps the space ${L_p}(\Omega , \mathcal {A}, P, B)$ into itself for each $p \geqslant 1$. The operator is linear, idempotent, constant preserving and contractive in ${L_p}$. For $B = {\mathbf {R}}$ and $p \ne 2$ these conditions characterize a conditional expectation operator. It turns out that in general these properties characterize conditional expectation operators for Banach-valued functions only for $p = 1$ and strictly convex Banach spaces.
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Additional Information
  • © Copyright 1981 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 81 (1981), 107-110
  • MSC: Primary 60B11; Secondary 60A05
  • DOI: https://doi.org/10.1090/S0002-9939-1981-0589148-7
  • MathSciNet review: 589148