limit of functions and the integration

Question

Let $f_n:\mathbb{R}\rightarrow\mathbb{R}$ be a sequence of non-negative Lebesgue measurable functions and suppose $\lim_{n\rightarrow \infty}\int_\mathbb{R}f_n=0.$ Then, must it be that $\lim_{n\rightarrow \infty}f_n(x)=0 $ almost everywhere? I can easily prove $\liminf_{n\rightarrow \infty}f_n(x)=0 $ almost everywhere by Fatou's lemma, but it seems somewhat hard to prove the original statement.

It is not true, since convergence in $L^1$ does not imply convergence a.e. . See here. — Eclipse Sun, Jul 01 '15 at 16:02

score 2 · Accepted Answer · answered Jul 01 '15 at 16:02

2

Consider the characteristic functions of $[0,1],[0,1/2],[1/2,1],[0,1/3],[1/3,2/3],[2/3,1], \dots $

answered Jul 01 '15 at 16:02

zhw.

105,693

1

This is called the typewriter sequence. – Julián Aguirre Jul 01 '15 at 16:10

limit of functions and the integration

1 Answers1