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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?

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    The proof is often omitted? What texts are you looking in? In my experience, this is usually one of the *first* things proved about independent random variables. See S. Resnick (1999), *A probability path*, Birkhauser, pp. 91ff, or R. Durrett, *Probability: Theory and Examples*, 4th. ed., Cambridge, sec 2.1 for details. The proof is a nice example of using the $\pi\mbox{-}\lambda$ theorem. Resnick's treatment is a bit more careful and detailed (or, tedious, depending on your perspective) than Durrett's, but they both follow along the very same lines.2011-09-15
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    Maybe I haven't checked enough text books ...2011-09-15
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    Which one(s) have you checked?2011-09-16
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    Here are some additional references on my shelf: **(1)** D. Williams (1991), *Probability with Martingales*, Cambridge, Ch. 4. **(2)** A. N. Shiryaev (1996), *Probability*, Springer, 2nd. ed., Ch. II, Sec. 5, pp. 179ff. **(3)** P. Billingsley (1995), *Probability and measure*, 3rd. ed, Wiley, Sec. 20, pp. 263ff. **(4)** J. Jacod and P. Protter (2004), *Probability essentials*, Springer, Ch. 10. In *every* one of the six examples given, the proof you asked for is the *very first* result given in the section on independent random variables.2011-09-18

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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 $\{(-\infty,x]|x\in\mathbb R\}$.