The Annals of Probability

Existence of Independent Complements in Regular Conditional Probability Spaces

D. Ramachandran

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Abstract

Let $(X, \mathscr{A}, P)$ be a probability space and $\mathscr{B}$ a sub-$\sigma$-algebra of $\mathscr{A}$. Some results on regular conditional probabilities given $\mathscr{B}$ are proved. Using these, when $\mathscr{A}$ is separable and $\mathscr{B}$ is a countably generated sub-$\sigma$-algebra of $\mathscr{A}$ such that there is a regular conditional probability given $\mathscr{B}$, necessary and sufficient conditions for the existence of an independent complement for $\mathscr{B}$ are given.

Article information

Source
Ann. Probab. Volume 7, Number 3 (1979), 433-443.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176995044

Digital Object Identifier
doi:10.1214/aop/1176995044

Mathematical Reviews number (MathSciNet)
MR528321

Zentralblatt MATH identifier
0399.28001

JSTOR
links.jstor.org

Subjects
Primary: 28A05: Classes of sets (Borel fields, $\sigma$-rings, etc.), measurable sets, Suslin sets, analytic sets [See also 03E15, 26A21, 54H05]
Secondary: 28A20: Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence 28A25: Integration with respect to measures and other set functions 28A35: Measures and integrals in product spaces

Keywords
Atoms of a $\sigma$-algebra separable $\sigma$-algebra continuous measure regular conditional probability measurable partial selector independent complement

Citation

Ramachandran, D. Existence of Independent Complements in Regular Conditional Probability Spaces. Ann. Probab. 7 (1979), no. 3, 433--443. doi:10.1214/aop/1176995044. https://projecteuclid.org/euclid.aop/1176995044


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