Proof of fact that $f \in L^r(X)$ iff $|f|^r \in L^1(X)$.

Question

I tried proof of this fact: $f \in L^r(X)$ iff $|f|^r \in L^1(X)$. Do you have an idea how it will proof of this fact?

Answer 1

I'm not sure what the confusion is: $$\|f\|_{L^r(X)}^r=\int\limits_X|f|^r\, d\mu=\int\limits_X\big||f|^r\big|\, d\mu=\||f|^r\|_{L^1(X)}.$$ It's just the definition of the two spaces. So, $\|f\|_{L^r}$ is finite if and only if $\| |f|^r\|_{L^1}$ is finite.