Suppose $f\in L^{p_1}(E)$ is bounded. Show that $f\in L^{p_2}(E)$ for all $p_2>p_1$.

Question

Suppose that $E$ is a measurable subset of $\mathbb{R}$ and $f$ is a bounded function in $L^{p_1}(E)$. Prove that $f$ is in $L^{p_2}(E)$ for all $p_2>p_1$.

I am confused about how to use the condition "$f$ is bounded" and also the case when the measure of $E$ is infinite.

Answer 1

Since $f$ is bounded, there is a constant $M>0$ such that $|f(x)|\le M$ for all $x\in E$. Then $$ |f|^{p_2}\le M^{p_2-p_1}|f|^{p_1}. $$