Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Let $\{f_n\}$ be a sequence of nonnegative functions dominated by some function $$ g \in L^1. $$ Then, the reverse Fatou lemma says $$ \limsup \int f_n \le \int \limsup f_n. $$

Is it possible to give an example where the inequality is strict?

I tried functions like $$ f_n=\chi_{(n,n+1)}, $$ but the dominating function is not integrble.

share|cite|improve this question
up vote 6 down vote accepted

$$f_{2n}=\chi_{[0,1]}\qquad f_{2n+1}=\chi_{[1,2]}\qquad g=\chi_{[0,2]}$$

share|cite|improve this answer

What about $f_n(x)=x(-1)^n \chi_{[-1,1]}(x)+\chi_{[-1,1]}, ~g=2\chi_{[-1,1]}$?

share|cite|improve this answer
nonnegative functions. – Did Apr 29 '13 at 16:58
sorry, but adding $1$ should fix that – Julian Apr 29 '13 at 17:01

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.