Suppose that $E[X^2Y^2] <\infty$. Can we conclude that $E[X^2]<\infty$?

Question

Suppose that $E[X^2Y^2] <\infty$. Can we conclude that $E[X^2]<\infty$, $E[Y^2]<\infty$ $E[|X|]<\infty$, $E[|Y|]<\infty$?

Attempt: If $X^2$ and $Y^2$ are uncorrelated, then $E[X^2Y^2]=E[X^2]E[Y^2]<\infty$ which implies $E[X^2]<\infty$ and $E[Y^2]<\infty$. One can conclude $E[X]<\infty$ and $E[Y]<\infty$ then $E[|X|]<\infty$, $E[|Y|]<\infty$.

Are there any other assumptions that will allow me to conclude all the implications.?

Answer 1

Here's a counterexample to all the claims: let $X, Y$ be defined on the common probability space $\Omega = [0, 1]$ (and Lebesgue measure) such that $$\begin{align*} X(\omega) &= \begin{cases} 1/ \omega, &0 < \omega < 1/2 \\ 0, & \text{otherwise}, \end{cases} \\ Y(\omega) &= \begin{cases} 1/ (1-\omega), &1/2 < \omega < 1 \\ 0, & \text{otherwise}. \end{cases} \end{align*} $$ Note that $XY(\omega) = 0$ for all $\omega$, but that $X, Y$ each have infinite expectation individually.

What makes this example work is that these variables are highly dependent on one another....