In basic probability, we know that if $A,B,C$ are events with $\mathbb{P}(C)>0$, then $$ \frac{\mathbb{P}(A\cap B\mid C)}{\mathbb{P}(B\mid C)} = \frac{\mathbb{P}(A\cap B\cap C)/\mathbb{P}(C)}{\mathbb{P}(B\cap C)/\mathbb{P}(C)} = \frac{\mathbb{P}(A\cap B\cap C)}{\mathbb{P}(B\cap C)} = \mathbb{P}(A\mid B\cap C). $$ I wonder if there is a measure-theoretic generalization of this formula? For example, if $\mathcal{C}$ is a $\sigma$-algebra, do we have something like $$ \mathbb{P}(A\cap B\mid\mathcal{C}) = \mathbb{P}(A\mid B\cap\mathcal{C})\mathbb{P}(B\mid\mathcal{C}) $$ where $B\cap\mathcal{C}=\{B\cap E:E\in\mathcal{C}\}$?
Edit: I suspect that this formula does not hold in general, because $\mathbb{P}(A\mid B\cap\mathcal{C})$ is $B\cap\mathcal{C}$-measurable and not assumed to be $\mathcal{C}$-measurable. I put this (probably incorrect) formula here only to illustrate what sort of generalization I am looking for, and because I cannot think of a better example.