Why is intersection of two independent set probability a multiplication process? Why is the probability of intersection of two independent sets $A$ and $B$, a multiplication of their respective probabilities i.e. Why is
$$\mathbb{P}(A \cap B) = \mathbb{P}(A) \cdot \mathbb{P}(B)?$$
this question is about the intuition behind the definition of independence of sets in a probability space

A proof is not necessary here, since this is a definition. If you want to understand this definition, consider an example. 
Observe all the girls of a small town. There are 1000 girls: 100 of them has a dog and 250 of them are blonde.
So now what is the probability of "a girl is blonde and has a dog" ? We assume here that the fact a girl is blonde is irrelevant to know if she has a dog or not. In term of probability, the events A "a girl is blonde" and B "a girl has a dog" are assumed independent. So, the 250 blond girls are just a subset of the 1000 girls. Since each one of them has the same probability to have a dog than every other girl (blonde or not) we can think that around 25 blond girls has a dog. 
In term of probability, $p(A \cap B) = 0.025 = 0.1 \times \frac{250}{1000}= p(A)p(B)$. Et voila !
Like diimension explains, we can also said that $p(B|A) = p(B)$. 
