Does $P(A \cap B) = 0$ imply that $A \cap B = \emptyset$? Given that $P(A \cap B) = 0$, where  $A$ and $B$ are two events, does this   imply that $A \cap B = \emptyset$ ?
Is it not possible to have a probability zero for an event which is not empty? 
Regards. 

No. For example, if I pick a random number from $[0,2]$, then $P([0,1]\cap [1,2])=0$, however, $[0,1]\cap[1,2]=\{1\}\neq \emptyset$.
