$L^q(X,\mu)\subset L^p(X,\mu)\iff X$ does not contain sets of arbitrarily large measure

Question

I'm trying to prove that if $(X,\mu)$ is a measure space then $L^q(X,\mu)\subset L^p(X,\mu)\iff X$ does not contain sets of arbitrarily large measure, for $1\leq p<q\leq \infty$. I've managed to prove that if $X$ does contain such sets, then the inclusion is false, but I'm having trouble proving the converse, that is, if there is $f\in L^q\backslash L^p$ then $X$ contains sets of arbitrarily large measure. I've tried to use a similiar argument as the one for proving that $L^p\not\subset L^q\implies X$ contains sets of arbitrarily small positive measure, by analyzing the sets $\{x\in X||f(x)|>n\}$, or in this case their complementary, but I can't guarantee their measure isn't infinite. Any suggestions?

Answer 1

If $X$ does not contain sets of arbitrarily large measure then $\mu (X) <\infty$. [There exits $M<\infty$ such that $\mu (E) \leq M$ for all $E$]. But the Holder's inequality shows that $(\int |f|^{p}d\mu)^{1/p}\leq (\int |f|^{q}d\mu)^{p/q}{(\mu (X)}^{1-\frac p q}$ which finishes the proof.