The Annals of Probability

A Note on the Strong Convergence of $\Sigma$-Algebras

Hirokichi Kudo

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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.

Article information

Source
Ann. Probab., Volume 2, Number 1 (1974), 76-83.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176996752

Digital Object Identifier
doi:10.1214/aop/1176996752

Mathematical Reviews number (MathSciNet)
MR370674

Zentralblatt MATH identifier
0275.60007

JSTOR
links.jstor.org

Subjects
Primary: 60G20: Generalized stochastic processes
Secondary: 60A05: Axioms; other general questions 28A05: Classes of sets (Borel fields, $\sigma$-rings, etc.), measurable sets, Suslin sets, analytic sets [See also 03E15, 26A21, 54H05]

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

Citation

Kudo, Hirokichi. A Note on the Strong Convergence of $\Sigma$-Algebras. Ann. Probab. 2 (1974), no. 1, 76--83. doi:10.1214/aop/1176996752. https://projecteuclid.org/euclid.aop/1176996752


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