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