I am having some trouble with the measure theoretic notion of conditional independence. In particular, it intuitively seems to me that if $\sigma(X)$ and $\sigma(Y)$ are conditionally independent with respect to $\sigma(Z)$, then it should be that for any $A \in \sigma(X)$, $B \in \sigma(Y)$, and $C \in \sigma(Z)$ we have $$P(A\cap B| C) = P(A|C)P(B|C).$$ Yet, I am having quite a bit of difficulty proving this statement.
Note: I take $\sigma(X)$ and $\sigma(Y)$ are conditionally independent with respect to $\sigma(Z)$ to mean that $X$ and $Y$ are conditionally independent on the $\sigma$-field, $\sigma(Z)$, so that for any $U \in \sigma(X)$, $V\in \sigma(Y)$, we have that $X$ and $Y$ are conditionally independent w.r.t. $\sigma(Z)$ if we have $P(U\cap V|\sigma(Z)) = P(U|\sigma(Z))P(V|\sigma(Z))$ $P$-a.s.