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February, 1974 A Note on the Strong Convergence of $\Sigma$-Algebras
Hirokichi Kudo
Ann. Probab. 2(1): 76-83 (February, 1974). DOI: 10.1214/aop/1176996752

Abstract

A quantity $\int |E\mathscr{B} f| dP$ (or equivalently $\int|u - P(A: \mathscr{B})| dP, 0 < u < 1)$ associated with a $\sigma$-algebra $\mathscr{B}$ is shown to act as a criterion for a type of convergence of $\sigma$-algebras. This quantity also defines an ordering of $\sigma$-algebras, so that upper and lower limits can be defined in terms of this quantity. Another criterion for the convergence of $\sigma$-algebras is described based on the existence of these limits.

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Hirokichi Kudo. "A Note on the Strong Convergence of $\Sigma$-Algebras." Ann. Probab. 2 (1) 76 - 83, February, 1974. https://doi.org/10.1214/aop/1176996752

Information

Published: February, 1974
First available in Project Euclid: 19 April 2007

zbMATH: 0275.60007
MathSciNet: MR370674
Digital Object Identifier: 10.1214/aop/1176996752

Subjects:
Primary: 60G20
Secondary: 28A05 , 60A05

Keywords: $L^1$-norm of conditional expectation , 60-00 , Existence of upper and lower limits of $\sigma$-algebras , Strong convergence of $\sigma$-algebras

Rights: Copyright © 1974 Institute of Mathematical Statistics

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Vol.2 • No. 1 • February, 1974
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