The product of integrable random variables need not be integrable
@Did gave a great example showing that in general the product
of two Lebesgue integrable functions need not be integrable. I thought
I could just tweak that example and prove this. However, I found the
“tweaking'' is not easy. Suppose random variable $X,Y\in L^{1}$.
It means that $E(X),E(Y)<+\infty$ , that is, $\int XdP,\int YdP<+\infty$.
I want to find an example where $\int XYdP$ is infinite.
The problem with the definition of expectation is that it is not convenient
to calculate. The convenient way of computing the expectation is to
use the density function : $E(X)=\int_{-\infty}^{\infty}xf(x)dx$
. However, if I use the density function, then unless $X$ and $Y$
are independent, contructing the density function for $XY$ and making
$E(XY)$ infinite will be tricky.
Any suggestion? Thank you.
Best Answer
Explaining the ingredients of the counterexample by Did: