# Definition of conditional expectation/independence

Conditional probability and conditional independence are unique almost surely, but relative to what: the conditioning field or the underlying field?

More precisely, consider the case of conditional independence. Let $\left(\Omega, \mathcal{A}, P\right)$ be a probability space, let $\mathcal{B}$ be a sub-$\sigma$-algebra of $\mathcal{A}$ and let $D,E\in\mathcal{A}$. Then by definition (see, e.g. Kallenberg (1995) p. 86) $D,E$ are conditionally independent given $\mathcal{B}$ iff $$P\left(\left.D\cap E\right|\mathcal{B}\right)=P\left(\left.D\right|\mathcal{B}\right)P\left(\left.E\right|\mathcal{B}\right)\space\space\mathrm{a.s}$$ But does "a.s." mean "up to a null set $F\in\mathcal{A}$" or "up to a null set $F\in\mathcal{B}$"?

-
I answered this on the other page. The answer is: BOTH, since $P(D\cap E\mid\mathcal B)$ and $P(D\mid\mathcal B)P(E\mid\mathcal B)$ are $\mathcal B$-measurable. –  Did Mar 10 '13 at 21:55
@Did: I see now. Thanks. –  Evan Aad Mar 10 '13 at 22:28