Independent events iff $P(A\cap B)=P(A)P(B)$ My teacher and wikipedia say that events $A$ and $B$ are independent if and only if $P(A\cap B)=P(A)P(B)$. But if one of $P(A)$ or $P(B)$ is 0, $P(A\cap B)=P(A) P(B)$ doesn't mean on does affect another. For example, pick a random number in $[0,1]$, let $A$ be the number is $0$, let $B$ be the number is in $[0,0.5)$. It is obvious $P(A)=0$ and $P(B)=0.5$. So $P(A \cap B)=0$. But if $A$ happens, $B$ must happen. Is it right that events $A$ and $B$ are independent if and only if $P(A\cap B)=P(A)P(B)$?
 A: The definition is correct in the case $P(A)=0$ (or $P(B)=0$), only if the event
$A$ (or $B$) is impossible.
As you have shown, the definition breaks down for events with $P(A)=0$, which can occur.
A: A mistake in your argument as far as I understand-
If you are picking a random number in a continues range of [0,1], the probability of getting an exact number is zero as the number of options is infinite, so your sample space is infinite. While it is mathematically correct, it makes no sense as such.
However, let's consider the conjecture A and B are independent if and only if P(A∩B)=P(A)*P(B).
If P(A) or P(B) is 0, then that it means it is an impossible event that will never occur, say P(Sun rises from the north). While let B be a possible event like the sun seemed orange in the evening today, so P(Sun was orange in the evening)!=0.
Now, P(A∩B) means that the sun should be both rising in the north and be orange in the evening, which is an impossibility as the sun cannot rise in the north, so naturally P(A∩B)=0, even though only P(A) was zero. 
A: Actually, that is the definition of independent. Then again, $P(B\mid A)$ is not $=1$ ("if $A$ happens, $B$ must happen"), it is not even defined.
