Proposition 12.1.12. Let $(X, \cm, \mu)$ be a measure space, $E$ be a normed vector space, $p \in [1, \infty]$, $\seq{f_n}\subset L^{p}(X; E)$, and $f \in L^{p}(X; E)$ such that $f_{n} \to f$ in $L^{p}$, then $f_{n} \to f$ in measure.
Proof. Let $\eps > 0$. If $p < \infty$, then by Markov’s Inequality,
\[\limv{n}\mu\bracs{|f_n - f| \ge \eps}\le \limv{n}\frac{1}{\eps^{p}}\norm{f_n - f}_{L^p(X; E)}^{p} = 0\]
If $p = \infty$, then there exists $N \in \natp$ such that $\norm{f_n - f}_{L^\infty(X; E)}< \eps$ for all $n \ge N$. In which case, $\mu\bracs{|f_n - f| \ge \eps}= 0$ for all $n \ge N$.$\square$