Probability of occurrence of 2 events simultaneously from 2 different sample spaces

Question

In my high school maths book there is written that the probability of occurrence of an event $A$ simultaneously with an event $B$ given that $A$ belongs to an experiment $X$ and $B$ belongs to an experiment $Y$ such that $X$ and $Y$ are independent experiments is given by $$ P(A \cap B)=P(A)·P(B) $$ The question is , how is $P(A \cap B)$ even defined given that $A$ and $B$ are from two different sample spaces? And why are we multiplying them, I mean what is the rule being followed here?

Answer 1

Let $(\Omega_X,2^{\Omega_X},\mathbb{P}_X)$ and $(\Omega_Y,2^{\Omega_Y},\mathbb{P}_Y)$, be the probability spaces corresponding to the experiments $X$ and $Y$, respectively. We can define their product probability space $(\Omega_X \times \Omega_Y,2^{\Omega_X\times \Omega_Y},\mathbb{P}_X\times \mathbb{P}_Y)$. Then the more precise way to write the statement in the book would be $$ (\mathbb{P}_X\times \mathbb{P}_Y)(A \times B) = \mathbb{P}_X(A)\cdot \mathbb{P}_Y(B)\quad \forall \;A \in 2^{\Omega_X},\, B \in 2^{\Omega_Y} $$ which of course is an overkill!