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The fatou's Lemma says

Let $ \left\{ f_n, n = 1,2,...\right\} $ be a sequence of non-negative measurable functions. Then $$ \liminf \int f_n \geq \int \liminf f_n$$

Some hint?

I think this Showing that the Fatou's lemma inequality can be strict. would work, but I don't understand why $\liminf f_n=0$ in the example he gives.

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$$f_n(x)=\begin{cases}\boldsymbol 1_{[0,1/2]}(x)&n\ \text{odd}\\ \boldsymbol 1_{[1/2,1]}(x)&n\ \text{even}\end{cases}$$

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