Is $fg\in L^1$ when $f,g\in L^1$?

Question

Let $(X,\mu)$ be a measure space. Let $f,g:X\rightarrow \mathbb{F}$ be functions in $L^1(\mu)$. ($\mathbb{F}$ is the field of real or complex numbers).

When I learned Riemann-Integral, it was quite immediate result that the integrability of $f,g$ implies the integrability of $fg$.

I'm studying 4 texts at a time now, but there's none about this in these books.

Is $fg$ integrable when $f,g$ are integrable? If not, what would be a counter example and under what condition does this hold?

Answer 1

Let $X=(0,1)$, $d\mu$ the Lebesgue measure and $$ f(x)=g(x)=x^{-1/2}. $$ Then $f,g\in L^1(X)$, but $(fg)=1/x\not\in L^1(X)$.