Is it true that a sequence $ f_n \to f $ of measurable functions is bounded by a norm of $ L^p $ then $ f_n $ converges to $f$ in $ L^p $?

Question

Is it true that a sequence $ f_n \to f $ of measurable functions is bounded by a norm of $ L^p $ then $ f_n $ converges to f in $ L^p $?

Is this true? If so, prove, if not, a counter example.

I just know that $ || f_n ||_{p} <M $ and that $\forall$ $\epsilon>0$ $\exists N_0 $ such that $\forall n>N_0 \Rightarrow |f_n -f |< \epsilon $

How to use this to show that

$\lim_{n \to \infty} ||f_n - f ||_{p} =0 $

It seems that in the linked question, we are asked to prove convergence in $L^1$, assuming that we work on a finite measurable space. This is not the same question, because we want convergence in $L^p$. — Davide Giraudo, Jun 23 '19 at 18:54

score 1 · Answer 1 · answered Jun 23 '19 at 17:16

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If we furthermore assume that $\lVert f_n\rVert_p\to \lVert f\rVert_p$, then it is true.

Otherwise, we may be in trouble, for example $f_n(x)=n^{1/p}\mathbf 1_{(0,1/n)}(x)$ where we endow $(0,1)$ with the Borel $\sigma$-algebra and Lebesgue measure.

answered Jun 23 '19 at 17:16

Davide Giraudo

172,925

A new picture? ${}$ – copper.hat Jun 23 '19 at 18:36
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@copper.hat Yes. – Davide Giraudo Jun 23 '19 at 18:53

Is it true that a sequence $ f_n \to f $ of measurable functions is bounded by a norm of $ L^p $ then $ f_n $ converges to $f$ in $ L^p $?

1 Answers1