I wish to show that for two random variables $X$ and $Y$, the condition $P(X\leq x, Y\leq ) = P(X\leq x)P(Y\leq y)$ implies that X and Y are independent.
I am approaching this problem from a measure theoretic perspective. So in particular I can write that $P(X\leq x) = \mu_X ((-\infty, x])$ and $P(Y\leq y) = \mu_Y ((-\infty, y])$. Also the independence condition here is that $\sigma(X)$ and $\sigma(Y)$ are independent, where these are the sigma algebras generated by the random variables.
Any help would be appreciated.