Bounded random variables $X,Y$ satisfying $\mathbb{E}(X^mY^n) = \mathbb{E}(X^m)\mathbb{E}(Y^n)$ for every $m, n\in\mathbb{N}$ are independent

Question

Suppose $X, Y$ are bounded random variables and we have that for every $m, n$ positive integers, $\mathbb{E}[X^mY^n] = \mathbb{E}[X^m]\mathbb{E}[Y^n]$. Then show that $X, Y$ are independent.

I have some idea of how this is supposed to go:

Using linearity, etc, $\mathbb{E}[f(X)g(Y)]$ = $\mathbb{E}[f(X)]\mathbb{E}[g(Y)]$ for all polynomials $f, g$.

After this we use limit theorems to extend this across all measurable functions, and thus characteristic functions, which will let us conclude that for every measurable set $A, B$, we have that $\mathbb{E}[1_{X \in A}\cdot 1_{Y \in B}] = \mathbb{E}[1_{X \in A}\cdot 1_{Y\in B}]$ which reduces to $P(X \in A, Y \in B) = P(X \in A)P(Y \in B)$ which gives independence.

But all the limit identities and how to use them has never been natural to me, so I wanted to verify and ask for help for the details.

First, for continuous functions $f, g$, we use the Stone-Weierstrass theorem to get polynomials sequences $f_n, g_n$ that converge uniformly to $f, g$ on the bounded interval that $X, Y$ belong to. Then $\mathbb{E}[f_n(X)] \rightarrow \mathbb{E}[f(X)]$, and similar for $Y, g$. Which theorem exactly are we using here for this convergence (assuming I'm not doing something completely wrong of course); dominated convergence theorem? Once we get this, we also then demonstrate that $\mathbb{E}[f_n(X)g_n(Y)] \rightarrow \mathbb{E}[f(X)g(Y)]$ which requires showing $f_ng_n \rightarrow fg$, but this just levarages standard analysis techniques and uses the uniform convergence, am I correct?

Next we want to extend this to all measurable functions $f, g$. For this we use the idea that there are sequences $f_n, g_n$ that will converge a.e pointwise to $f, g$. Is this enough to conclude that $\mathbb{E}[f_n(X)] \rightarrow \mathbb{E}[f(X)]$, and similar for $Y, g$? It seems we need stronger assumptions here. I'm not entirely sure how valid this last step is and would appreciate some detail.

Answer 1

Your strategy almost works, except that not every simple function is the pointwise limit of continuous functions (e.g., $1_{\mathbb{Q}\cap[0,1]}$ since it has Baire class 2), so you need to find a way around that. The usual way (asked before many times, see e.g. here, here and here) is to go via the characteristic function $\varphi_{\vec{X}}(\vec\theta):=\mathbb{E}[\exp(i\vec{\theta}\cdot\vec{X})]$ or its cousin the moment generating function, but let me describe a more measure-theoretic way with the monotone class theorem.

Let's start from the beginning. We could transfer the problem to $\Omega=[-M,M]^2$ and $X,Y$ being the coordinate function without any loss of generality.

Stone-Weierstrass and uniform convergence: if $X_n\to X$ uniformly, then $\mathbb{E}X_n\to\mathbb{E}X$ since $\mathbb{E}\lvert X_n-X\rvert\leq\sup\lvert X_n-X\rvert\to 0$. In particular, if $f,g$ are continuous, then there exists $f_n,g_n$ polynomials converging uniformly to $f,g$. Then $(x,y)\mapsto f_n(x)g_n(y)\to f(x)g(y)$ uniformly and thus $\mathbb{E}[f_n(X)g_n(Y)]\to\mathbb{E}[f(X)g(Y)]$. Hence $\mathbb{E}[f(X)g(Y)]=\mathbb{E}[f(X)]\mathbb{E}[g(Y)]$ for all continuous $f,g$.

Then we appeal to

Monotone Class Theorem (for Functions) Let $\mathcal{K}$ be a collection of bounded $\mathbb{R}$-valued functions on $\Omega$ closed under multiplication, and $\mathcal{B}=\sigma(\mathcal{K})$. Suppose $\mathcal{H}\supseteq\mathcal{K}$ is a vector space over $\mathbb{R}$ of bounded $\mathbb{R}$-valued functions containing $1_\Omega$ and closed under uniform limit of nonnegative functions nondecreasing to a bounded function. Then $\mathcal{H}$ contains $\mathcal{B}_b$, the set of all bounded $\mathcal{B}$-measurable $\mathbb{R}$-valued functions.

Here we use $\mathcal{K}_1$ is our polynomials on $[-M,M]$ and $\mathcal{H}_1$ the set of all functions $f$ such that $\mathbb{E}[f(X)g(Y)]=\mathbb{E}[f(X)]\mathbb{E}[g(Y)]$ for all $g\in\mathcal{K}_1$. By MCT, $\mathcal{H}_1$ contains all bounded measurable functions on $[-M,M]$. Now MCT again, with $\mathcal{K}_2=\mathcal{B}_b$ and $\mathcal{H}_2$ being those $g$ satisfying $\mathbb{E}[f(X)g(Y)]=\mathbb{E}[f(X)]\mathbb{E}[g(Y)]$ for all $f\in\mathcal{B}_b$ gives all $g\in\mathcal{B}_b$ too.