Proving that if $P(A \cap B) = 0$, then A and B are mutually exclusive I'm trying to prove that if $P(A \cap B) = 0$, then A and B must be mutually exclusive sets. This seems a relatively simple problem considering it is required by definition for $P(A \cap B)$ to be equal to $0$ in order for $A$ and $B$ to be exclusive. My problem arises when I try and account for the infinitesimally small dart problem. That is, taking an infinitely small dart and calculating the probability that it will land on any real number in some given range. For example, 
Let $A = [0, .5]$
Let $B = [.5, 1]$
Since there are an infinite number of real numbers between 0 and 1, the probability of it landing on any number, (including in this case the intersection $\{.5\}$) is $0$.  So,  $P(A \cap B) = 0$ but the two sets are not mutually exclusive. How can I reconcile this?

If $A,B$ are disjoint sets, then $\mathsf P(A\cap B)=0$.   However the converse cannot be claimed.
$\mathsf P(A\cap B)=0$ merely states that $A\cap B$ is a null set under the probability measure.   It is not a strong enough claim to infer emptiness of the intersection.
