TITLE: The definition of independent discrete random variables
QUESTION [7 upvotes]: In probability books, the definition of independent discrete random variables are often given as

The random variables $X$ and $Y$ are said to be independent if
  $\mathbb P(X \leq x, Y \leq y) = \mathbb P(X \leq x) \mathbb P(Y \leq y)$
  for any two real numbers $x$ and $y$, where $\mathbb P(X \leq x, Y \leq y)$ represents the
  probability of occurrence of both event $\{X \leq x\}$ and event $\{Y \leq y\}$.

or

$\mathbb P(X \in A, Y \in B) = \mathbb P(X \in A) \mathbb P(Y \in B)$

And the 2 definitions are alleged to be identical.  But the proof is often omitted.  Although it's intuitively correct, I still want to see a proof.  Could anyone show me how to prove this?

REPLY [2 votes]: These two definitions are equivalent due to the following reason. I assume that you mean $A,B$ be Borel measurable. Then the 2nd definition says that $\sigma(X)$ is independent of $\sigma(Y)$ where
$$
\sigma(X) = \{X^{-1}(B)|B\in\mathcal{B}(\mathbb{R})\}
$$
and
$$
\sigma(Y) = \{Y^{-1}(B)|B\in\mathcal{B}(\mathbb{R})\}.
$$
Clearly, 2nd definitino implies the first one. To see that the first implies the second just recall that $\mathcal{B}(\mathbb R)$ is the smallest $\sigma$-algebra which includes the class $\{(-\infty,x]|x\in\mathbb R\}$.