Is product of Lebesgue integrable functions also Lebesgue integrable

Question

If $ f, g \in L(I), $ does $ fg \in L(I)$?

I tried to solve this that way:

$f$ and $g$ are Lebesgue integrable for each function there is a sequence of step function ${f_n}$ and ${g_n}$ that satisfy:

• ${f_n}$ and ${g_n}$ are increasing, bounded and $ f_n \rightarrow f , g_n \rightarrow g $
• $ \int_{I} f_n, \int_{I} g_n $ is bounded

so the sequence $\{f_n g_n\}$ satisfy those conditions and therefore $fg \in L(I)$

Am I right?

thanks

Answer 1

Assuming that $I$ is an interval of $\mathbb R$, then the answer is negative. Take $I=(0,1]$, and $f(x)=g(x)=\frac1{\sqrt x}$.