Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I would like to prove the following:

If $f_n \geq 0$ and $f_n\rightarrow f$ in measure, then $\int f d\mu \leq \liminf \int f_n d\mu$.

So far I have

Since $f_n \rightarrow f$ in measure, then there exists a subsequence $\left\{ f_{n_j}\right\}$ that converges to $f$ a.e. Then $\displaystyle \liminf_{n\rightarrow \infty} f_n(x) \leq f(x)$ a.e.

I want to be able to apply Fatou's Lemma (I say that because it looks a lot like Fatou's Lemma). Could someone show me how to finish proving this? I'm an undergraduate student and I am studying measure theory on my own for the first time, so it would be nice to see a good proof of this. Thank you!!

share|improve this question

1 Answer 1

If the result did not hold, then there is a subsequence $n_k$ such that $\int f d\mu > \lim \int f_{n_k}d\mu$. From this subsequence, extract a further subsequence which converges almost everywhere. Apply Fatou to this subsequence to arrive at a contradiction.

share|improve this answer

Your Answer

 
discard

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.