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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!!

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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.

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