Lebesgue integral iteration with the limit

Question

For every n, let $f_{n}:[0;1]\rightarrow{\mathbb R}$ a measurable Lebesgue function: $\int_{[0;1]} {\left| f_{n}\right\| ^{2}}\mathrm{d}m\leq{5} $

Is true that $\lim\limits_{n\to\infty} \int_{[0;1]} {\left| f_{n}\right\| ^{2}}\mathrm{d}m=0$? Is true that $\lim\limits_{n\to\infty} \int_{[0;1]} {\left| f_{n}\right\|}\mathrm{d}m=0$?

Welcome to math SE. What have you tried so far? – Alain Remillard Dec 07 '20 at 15:42 — Alain Remillard, Dec 07 '20 at 15:42

score 2 · Answer 1 · answered Dec 05 '20 at 23:39

2

No.

Consider $f_n(x) = \sqrt{5}$ for all $x$ for all $n$.

Then $\int |f_n|^2 \mathrm{d}m \leq 5$, but both $\lim \int |f_n|^2 \mathrm{d}m$ and $\lim \int |f_n| \mathrm{d}m$ are nonzero.

I hope this helps ^_^

answered Dec 05 '20 at 23:39

HallaSurvivor

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Thank you. So useful! – user857880 Dec 06 '20 at 12:49
And if we added the hypothesis $f_{n}(x)\rightarrow{0}$ when $n\rightarrow{\infty}$ for every almost $x\in [0,1]$, would it be true? – user857880 Dec 06 '20 at 13:14
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@user857880 Please consider accepting a correct answer. – WhatsUp Mar 18 '21 at 23:18

Lebesgue integral iteration with the limit

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