Is it true that, given a space $X$ and a probability measure $\mu$ on it, given some sets $A, B \subset X$ and a finite number of disjoint sets $C_{\sigma}$ such that $\bigcup_{\sigma} C_{\sigma} =X$, the following inequality holds,

$$\mu (A \cap B ) = \sum_{\sigma}\mu(A \cap C_{\sigma}\cap B) \leq \sum_{\sigma} \frac{ \mu(A \cap C_{\sigma} ) } {\mu(A)} \frac{\mu( C_{\sigma} \cap B)}{\mu(C_{\sigma})}, ? $$

Probabily it is wrong in general ( but I am not sure ), but is it true that if $A$ and $B$ are both equal to the union of a finite number of sets $C_{\sigma}$, than the equality holds? I think yes!