Bounded functions and inclusion on the $L^{p} $ spaces

Question

Let $f \in L^p(X) $ be a bounded function, show that $\forall q \geq p, \; f \in L^q(X)$

I'm aware that this result holds for the case where $\mu(X)<\infty$ but I can't see how to prove this for when the measure of the whole space is not necessarily finite. I've tried using Holder's inequality but that didn't seem to help much.

Answer 1

Since $f$ is bounded there is some $K >0$ such that $$ |f| \leq K. $$ Hence $$ \int |f|^q d\mu= \int |f|^p|f|^{q-p} d\mu \leq \int K^{q-p}|f|^p d\mu=K^{q-p} \int |f|^p d\mu. $$ Since $f \in L_p$, $\int |f|^p d\mu$ is finite. Hence $\int |f|^q d\mu$ is finite and $f \in L_q$.