There is a common attitude in the text books on probability that the so-called product rule is an obvious property, when events are independent, i.e., $P(A\cap B)=P(A)P(B)$ when $A$ and $B$ are independent events. Yet, this is NOT an axiom that a probability must satisfy, nor it is a property that follows from the axioms. That is because there is no definition of the concept of independent events in the axioms.
Of course, the product rule is the actual definition of the (stochastic) independence of two events, if one wants to evade the concept of conditional probability. So you, clearly, see the chicken-and-the-egg conundrum.
Still, we continue to use the rule because we intuitively feel the independence property is true, which seems natural in those cases when our random experiment is obtained as a succession of two (simpler) random experiments which are causal independent.
And now my question: how is this idea (of causal independence) formalized in the context of the axiomatic approach to probabilities such that we can prove from here the product rule?
$(|)=(),(|)=\frac{(∩)}{()}→(∩)=(|)()→(∩)=()()$