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.
Publié le : 1979-06-14
Classification:
Atoms of a $\sigma$-algebra,
separable $\sigma$-algebra,
continuous measure,
regular conditional probability,
measurable partial selector,
independent complement,
28A05,
28A20,
28A25,
28A35
@article{1176995044,
author = {Ramachandran, D.},
title = {Existence of Independent Complements in Regular Conditional Probability Spaces},
journal = {Ann. Probab.},
volume = {7},
number = {6},
year = {1979},
pages = { 433-443},
language = {en},
url = {http://dml.mathdoc.fr/item/1176995044}
}
Ramachandran, D. Existence of Independent Complements in Regular Conditional Probability Spaces. Ann. Probab., Tome 7 (1979) no. 6, pp. 433-443. http://gdmltest.u-ga.fr/item/1176995044/