Lp inclusion on a finite measure space
= Lp inclusion on a finite measure space
On a finite measure space, $L^q\subseteq L^p$ for $0<p<q\leq\infty$. For $1\leq p<q<\infty$, <Holder inequality>[Hölder's inequality] gives
$$
\lVert f\rVert_p
\leq\mu(X)^{1/p-1/q}\lVert f\rVert_q,
$$
with the same formula interpreted as $\lVert f\rVert_p\leq\mu(X)^{1/p}\lVert f\rVert_\infty$ when $q=\infty$.