Expectation of product of independent random variables

Question

I'm stuck trying to show $E(XY) = E(X)E(Y)$ for $X, Y$ nonnegative bounded independent random variables on a probability space. The definition of independence is that $P(\{ X \in B\} \cap \{ Y \in C\}) = P(X \in B) P(Y \in C)$ for Borel sets $B$ and $C$. I'm not assuming $X$ or $Y$ have probability density functions so I cannot use them. Nor can I use conditional expectation.

Answer 1

If two random variables $X,Y$ have a joint distribution then they are independent if and only if the corresponding CDF's satisfy: $$F_{X,Y}(x,y)=F_X(x)F_Y(y)\tag1$$ Here $(1)$ is a necessary but also sufficient condition for:$$\mathsf P_{X,Y}=\mathsf P_X\times \mathsf P_Y$$where $\mathsf P_{X,Y}$ denotes the probability on $(\mathbb R^2,\mathcal B^2)$ induced by $(X,Y):\Omega\to\mathbb R^2$ and $\mathsf P_X,\mathsf P_Y$ denote the probabilities on $(\mathbb R,\mathcal B)$ induced by $X:\Omega\to\mathbb R$ and $Y:\Omega\to\mathbb R$.

Then under suitable conditions: $$\mathsf EXY=\int xydF_{X,Y}(x,y)=\int\int xydF_X(x)dF_Y(y)=\int xdF_X(x)\int ydF_Y(y)=\mathsf EX\mathsf EY$$

Answer 2

Here are some hints for this classical result:

Start with $X = 1_A$, $Y = 1_B$, with $A$ and $B$ borel sets.

Then, use the fact that any positive random variable $X$ can be written as : $X = \sum_{k\geq 0}{b_k 1_{B_k}}$ with $b_k$ being some positive real numbers and $B_k$ borel sets. Prove the equality for any positive random variables $X$ and $Y$.

Finally write $X = X_+ - X_-$, $Y = Y_+ - Y_-$ and conclude.