Independence of an event with null probability with another event. Suppose we have an event $A$ with $P(A) = 0$, and let $B$ be any other event. Prove that $A$ and $B$ are independent.
It seems rather obvious that since $P(A \cap B) = P(A)P(B)$ if they're independent, and $P(A)P(B) = 0$, so $A$ and $B$ will be independent regardless since $A \cap B$ must have Lebesgue measure 0. However, I'm not satisfied with this "proof". Is it possible that it's not always true?

It is always true. If $\Pr(A)=0$, then $\Pr(A\cap B)=0$. So $\Pr(A\cap B)=\Pr(A)\Pr(B)$. It is hardly ever Lebesgue measure, it is whatever probability measure in involved.  
