Lp inclusion on a finite measure space

On a finite measure space, $L^q\subseteq L^p$ for $0<p<q\le \infty$. For $1\le p<q<\infty$, Hölder's inequality giveswith the same formula interpreted as $\Vert f{\Vert }_p\le \mu (X{)}^{1/p}\Vert f{\Vert }_{\infty }$ when $q=\infty$.

If $f_n\to f$ in $L^p$ for $1\le p<\infty$, choose a subsequence with $\Vert f_{n_k}-f{\Vert }_p^p\le 2^{-k}$. Tonelli's theorem makes ${\sum }_k|f_{n_k}-f{|}^p$ finite almost everywhere, so $f_{n_k}\to f$ almost everywhere. For $p=\infty$, norm convergence itself gives almost-everywhere convergence after removing one common null set.

Let $X\subseteq {\mathbb{R}}^n$ be Borel. If $L^q(X)\subseteq L^p(X)$ for $1\le p<q\le \infty$, the closed graph theorem makes the inclusion continuous. Applying its norm bound to indicators of $X\cap B(0,R)$ shows that their measures are uniformly bounded, hence $\mu (X)<\infty$.

The essential supremum of a measurable function is the least number $M$ such that $f\le M$ almost everywhere. It equals the $L^{\infty }$ norm for a nonnegative function and, on a finite measure space, $\Vert f{\Vert }_p\to \Vert f{\Vert }_{\infty }$ as $p\to \infty$.

On $(0,1)$, $x^{-a}$ is integrable exactly when $a<1$, providing sharp counterexamples between finite-measure $L^p$ spaces.