Convergence in Norm Implies Convergence in Measure

Theorem
Let $\struct {X, \Sigma, \mu}$ be a measure space, and let $p \in \R, p \ge 1$.

Let $\sequence {f_n}_{n \mathop \in \N}, f_n : X \to \R$ be a sequence of $p$-integrable functions.

Also, let $f: X \to \R$ be a $p$-integrable function.

Suppose that $f_n$ converges in norm to $f$ (in the $L^p$-norm).

Then $f_n$ converges in measure to $f$ (in $\mu$).

That is:


 * $\ds \operatorname {\LL^{\textit p}-\!\lim\,} \limits_{n \mathop \to \infty} f_n = f \implies \operatorname {\mu-\!\lim\,} \limits_{n \mathop \to \infty} f_n = f$

Proof
Let $\epsilon > 0$.

Then: