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On Products of Conditional Expectation Operators

Published online by Cambridge University Press:  20 November 2018

Radu Zaharopol
Radu Zaharopol*
Affiliation:
School of Mathematics The Institute for Advanced Study Princeton, NJ 08540
*
Current address: SUNY at Binghampton Binghampton, NY., 13901
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Abstract

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Let (X, Σ, μ) be a probability space, let f1, f2, ..., Fk be k σ-subalgebras of Σ, and let p ∊ R be such that 1 < p < + ∞. Let Pi :LP(X, Σ, μ) → LP(X, Σ, μ) be the conditional expectation operator corresponding to fi for every i = 1,2,…, k, and set T = P1 . . . Pk. Our goal in the note is to give a new and simpler proof of the fact that for every f ∊ LP(X, Σ, μ), the sequence (Tnf)n∊N converges in the norm topology of LP(X, Σ, μ), and that its limit is the conditional expectation of f with respect to f1 ∩ f2 ∩ … ∩ Fk.

Keywords

47A3528D99
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 33 , Issue 3 , 01 September 1990 , pp. 257 - 260
Copyright
Copyright © Canadian Mathematical Society 1990

Footnotes

This work was done while the author was supported by a grant-in-aid from The Institute for Advanced Study. The grant-in-aid is part of a grant made to The Institute by the National Science Foundation. The author would like to express his deep gratitude to all the people who made his stay at The Institute possible.

I am indebted to Professor Harry Furstenberg for bringing to my attention the topic discussed in this note, and to Professor Mustafa A. Akcoglu, Professor Tsuyoshi Ando, Professor Gian-Carlo Rota and Professor Louis Sucheston for many instructive comments which were very useful to my work on the paper.

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