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I'm trying to use an example to show that Fatou's lemma can not be strengthened to equality. I was given a hint, which I'm not quite sure how to use. I was told that if I look at the one-dimensional case, and let $f_k(x)=\begin{cases} k, &\quad\text{if } - \frac{1}{k} \leq x \leq \frac{1}{k}\\ 0, &\quad\text{elsewhere} \ \end{cases}$ , then $\int f_kdm=\frac{2}{k}(k)=2, \forall k$, and $g_k(x) \to 0, \forall x$, except for $x=0$, for which $g_k \to \infty$. How can I use this to show that equality can not be achieved? I thought specific examples couldn't be used to prove general behaviors? Can someone please help?

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    @Julián Aguirre, so basically if I just show that this specific example leads to strict inequality, then I've achieved my goal?2012-11-19

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