If $f$ and $g$ be integrable functions and real-valued on $(X,M,\mu)$ , which assertion is correct?

Question

If $f$ and $g$ are integrable functions and real-value on $(X,M,\mu)$, which assertion is correct?

1-$fg\in L^1(\mu)$

2-$fg\in L^2(\mu)$

3-No.

Answer 1

The pointwise product is not $L^1$ stable in general. For instance, take $f(x)=g(x)=\frac{1}{\sqrt{x}}$ on $X=(0,1)$ equipped with the Lebesgue measure. Then $f$ is $L^1$, but $f^2$ is not.

The good product to consider to make $L^1$ an algebra is the convolution product $$ (f*g)(x):=\int_Xf(t)g(x-t)d\mu(t). $$ Then for every $f,g\in L^1$, the function $f*g$ is measurable and $\|f*g\|_1\ \leq \|f\|_1\|g\|_1$, as shown by an application of Fubini. This turns $L^1$ into a Banach algebra.