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 measurable functions on $ [0,1]$ such that $\int_{0}^1 |f_n|^2 dm \le C$.

Assume that there exist a function $f$ such that $\int_{0}^1 |f_n-f|dm \to 0$ as $n \to \infty $

show that $\int_{0}^1 |f|^2 dm \le C$ and it is true that $\int_{0}^1 |f_n-f|^2dm \to 0$ as $n \to \infty $ ?

I have no idea how to begin, any hints to start me off would be appreciated. Thanks

share|cite|improve this question
The conclusion is false. Please check if your question is correctly formulated. – 23rd Dec 8 '12 at 16:32
@ richard, got the mistake, thanks – jake Dec 8 '12 at 17:10
You are welcome! – 23rd Dec 8 '12 at 17:22
up vote 2 down vote accepted

Begin extracting a subsequence $\{f_{n_k}\}$ such that for all $k$, $$\int_{[0,1]}|f_{n_k}-f|dm< 4^{-k},$$ then that the sequence $\{f_{n_k}\}$ converges almost everywhere to $f$.

The first assertion is a consequence of Fatou's lemma.

The second one is not true: take $f_n:=\sqrt n\chi_{(0,n^{-1})}$, whose $L^2$ norm is $1$, and $f=0$. Then $\int_{(0,1)}f_n=n^{-1/2}\to 0$.

share|cite|improve this answer

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.