= Solution
For $1\leq p<\infty$, the <Lebesgue space> $L^p(X)$ consists of equivalence classes modulo equality almost everywhere of measurable functions satisfying
$$
\lVert f\rVert_p
=\left(\int_X|f|^p\,d\mu\right)^{1/p}<\infty.
$$
The space $L^\infty(X)$ consists of essentially bounded measurable functions modulo equality almost everywhere, with norm
$$
\lVert f\rVert_\infty=\operatorname*{ess\,sup}_{x\in X}|f(x)|.
$$
Let $1\leq p<q<\infty$ and put $r=q/p>1$. By <Holder inequality>[Hölder's inequality],
$$
\int_X|f|^p
\leq
\left(\int_X|f|^{pr}\right)^{1/r}
\mu(X)^{1-1/r}.
$$
Taking $p$th roots gives
$$
\boxed{\lVert f\rVert_p
\leq\mu(X)^{1/p-1/q}\lVert f\rVert_q.}
$$
For $q=\infty$, directly integrating $|f|^p\leq\lVert f\rVert_\infty^p$ gives the same endpoint estimate. Hence $L^q(X)\subseteq L^p(X)$ when $\mu(X)<\infty$, as in <Lp inclusion on a finite measure space>.
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