## The Annals of Probability

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

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

#### 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