Combinations of integrable functions

Question

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

1. $fg\in L^1 (\mu)$
2. $fg\in L^2 (\mu)$
3. $\sqrt{f^2 +g^2}\in L^1 (\mu)$
4. None of the above.

Answer 1

The correct answer is $3$. Indeed, $f^2+g^2\leq f^2+2|f||g|+g^2=(|f|+|g|)^2$ so $$ \int_X \sqrt{f^2+g^2}d\mu\leq \int_X (|f|+|g|)d\mu=\int_X|f|d\mu+\int_X|g|d\mu<\infty. $$ To see that $1$ and $2$ fail, consider $f(x)=g(x)=\frac{1}{\sqrt{x}}$ in $L^1((0,1))$ for instance.

Answer 2

The only correct assertion is no. 3.

As a counterexample for 1 and 2, take $(X,M,\mu)$ as $[0,1]\subset\mathbb R$ with the Lebesgue measure, and $f(x)=g(x)=\frac{1}{\sqrt x}$.

No. 3. works well, indeed $\sqrt{f^2+g^2}\leq |f|+|g|$ as you can easily see by squaring both sides.