Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 3 / 22F / a / Solution

For $1\le p<\infty$, the Lebesgue space $L^p(X)$ consists of equivalence classes modulo equality almost everywhere of measurable functions satisfyingThe space $L^{\infty }(X)$ consists of essentially bounded measurable functions modulo equality almost everywhere, with norm

Let $1\le p<q<\infty$ and put $r=q/p>1$. By Hölder's inequality,Taking $p$th roots givesFor $q=\infty$, directly integrating $|f{|}^p\le \Vert f{\Vert }_{\infty }^p$ 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.