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.
Citation
D. Ramachandran. "Existence of Independent Complements in Regular Conditional Probability Spaces." Ann. Probab. 7 (3) 433 - 443, June, 1979. https://doi.org/10.1214/aop/1176995044
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