Let $X$ be a compact space and $\mu$ the Lebesgue measure, with $\mu(X)=1$. Let A and B be two subsets of $X$ with positive measure. What can I say about the relations between $\mu (A \cap B)$ and $\mu (A) \mu(B)$? Is one of the two quantities always larger of equal to the other one?
If not, how can I prove that, given a finite number of disjoint sets $C_\sigma$ such that $\bigcup_{\sigma} C_{\sigma}= X$, then
$\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})}, ? $