## The Annals of Probability

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

### Existence of Independent Complements in Regular Conditional Probability Spaces

#### 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