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

If $\left\{f_n\right\}$ are uniformly integrable and $f_n\overset{a.e.}{\rightarrow}f$ ($f$ measurable), is $f$ integrable? Can "uniformly integrable" be weakened to "integrable"?

share|cite|improve this question

Put $f_n(x) = n(n+1)I_{(1/n+1, 1/n)}$, $x\in[0,1]$. Then $f_n\to 0$ in $[0,1]$ and the sequence of the $f_n$ is $L^1$-bounded. However, $f_n\not\rightarrow 0$ in $L^1$ as $n\to\infty$, since $\|f_n\|_1 = 1$ for all $n$.

share|cite|improve this answer
Thanks. I can only mark one answer as correct so i'm going to mark my own, since this was the question i was principally interested in. – Evan Aad Aug 9 '12 at 11:55
But this is an example of some integrable functions $f_n$ such that $f_n\to f$ almost everywhere and $f$ is integrable. Hence not an answer to the question asked. – Did Aug 9 '12 at 13:08
up vote 0 down vote accepted
  1. Yes

    $\int \left|f\right|\overset{\mathrm{a}}{\leq}\liminf\int f_n\overset{\mathrm{b}}{<}\infty$

    a. Fatou's lemma

    b. Uniform integrability $\implies$ $\sup \int f_n<\infty$ (e.g. Klenke, Theorem 6.24i)

  2. No, e.g. $f_n:=\mathbb{1}_{\left[-n,n\right]}$ (borrowed from Per Manne's comment below)

share|cite|improve this answer
The answer to the second question is no. If $f_n(x)$ is the indicator function of the interval $[0,n]$ then each $f_n$ is integrable, but the limit is not. – Per Manne Aug 9 '12 at 11:39
@PerManne: Thanks. – Evan Aad Aug 9 '12 at 11:54
what is definition of uniformly integrable? – nim Nov 29 '13 at 10:55
Can we strengthen the result to say that the limit of uniformly integrable functions is uniformly integrable (as opposed to just integrable)? – QuantumDots Nov 16 '15 at 0:52

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.