Can we define independence without using conditional probability? A textbook I'm currently reading defines independence as follows:
Two events $A$ and $B$ are independent if $P(A|B)=P(A)$ (provided that the probability of the events are positive)
Then derives the following as a theorem:
Two events $A$ and $B$ are independent iff $P(A \cap B)=P(A)P(B)$
My question is this: can we give this theorem as a definition, that is, if we don't have the concept of conditional probability, can we still define independence as follows:
Two events $A$ and $B$ are independent if $P(A \cap B)=P(A)P(B)$
 A: As a counterpoint to Matthew Drury's answer, I prefer the second definition $$\text{Events}~A ~\text{and}~ B~\text{are said to be independent if and only if}~ P(A\cap B) = P(A)P(B)$$ over the first because it avoids the asymmetry in the first definition where $B$ can be "independent of" $A$ because $P(B\mid A) = P(B)$ holds while $A$ cannot be said to be "independent of" $B$ because $P(B) = 0$ and so $P(A\mid B)$ is undefined.  Yes, the definition in terms of conditional probabilities is more intuitive (when it works) but to my mind, independence is a fundamental concept that should not (and does not) need the notion of conditional probability to define.
A: You can, as the two equations are equivalent (at least in the cases where $P(B)$ is not zero).
On the other hand, I prefer the first as a definition, as it communicates the intent of introducing the concept.  The equations
$$ P(A \mid B) = P(A) $$
directly expresses that knowledge of $B$ does not influence our state of knowledge about $A$, which, in a sense, is what independence actually means.
