Convergence in Norm Implies Convergence in Measure

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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 $p$-norm).


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

That is:

$\displaystyle \operatorname {\mathcal L^{\textit p}-\!\lim\,} \limits_{n \mathop \to \infty} f_n = f \implies \operatorname {\mu-\!\lim\,} \limits_{n \mathop \to \infty} f_n = f$


Proof


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