Why is $fg$ integrable w.r.t. a probability measure if $f,g$ are Lebesgue integrable?

Question

In one of the proofs, my text mentions that if $f,g$ are Lebesgue integrable then $fg$ is integrable with respect to a probability measure.

I guess I have missed something, since it doesn't look obvious to me. Any hints are greatly appreciated.

Edit. Here is the context:

My text claims that

If $X, Y$ are independent and integrable r.v., $f,g$ are measurable and integrable (with respect to Lebesgue I assumed), then $\mathbb E(f(X)g(Y))=\mathbb E(f(X))\mathbb E(g(Y))$.

And then a sketch of the proof is given: from measurability of $f,g$ and independence of $X, Y$ trivially follows independence of $f(X), g(Y)$. Now use Fubini to show the equality. The use of Fubini is justified since integrability of $fg$ with respect to $\mathbb P_Z$, where $Z:=(X,Y)$, follows from integrability of $f$ and $g$.

Answer 1

Key hypothesis: (stochastic) independence.

This can be surprising at first but indeed, if $n$ random variables $U_i$ are integrable and independent then their product $U=U_1U_2\cdots U_n$ is integrable as well and furthermore $E(U)$ is the product $E(U_1)E(U_2)\cdots E(U_n)$.

In your context, $U_1=f(X)$ and $U_2=g(Y)$ are independent as soon as $X$ and $Y$ are independent hence, if $f(X)$ and $g(Y)$ are integrable then $U=f(X)g(Y)$ is integrable as well. A correct formulation of the result you cite would be as follows.

If $X$ and $Y$ are independent random variables and $f$ and $g$ are measurable functions, integrable with respect to $P_X$ and $P_Y$ respectively, then the random variable $f(X)g(Y)$ is integrable and $E(f(X)g(Y))=E(f(X))\cdot E(g(Y))$.

Note that $f$ and $g$ are not assumed to be integrable with respect to Lebesgue measure but with respect to the probability measures called the distribution of $X$ and the distribution of $Y$. Thus, the result could be rephrased as the fact that, as soon as $f$ is integrable with respect to $P_X$ and $g$ is integrable with respect to $P_Y$, the function $f\otimes g$, defined by $(f\otimes g)(x,y)=f(x)g(y)$ for every suitable $(x,y)$, is integrable with respect to the product $P_X\otimes P_Y$.