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I'm studying measure theory for the first time, and I just came across Fatou's Lemma.

Why isn't it true that for any sequence of functions $\left\{ f_n \right\}$ in $L^+$ we always have that $$\int \displaystyle \liminf_{n\rightarrow \infty} f_n d\mu =\liminf_{n\rightarrow \infty} \int f_n d\mu\ ?$$

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  • $\begingroup$ Since you have figured this out, it is also useful to know the form of Fatou's Lemma $$\int_E f \leq \text{liminf} \int_E f_n $$ $\endgroup$ – user1876508 Nov 1 '13 at 6:43
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consider the sequence of functions$$f_n = n 1_{(0,\frac{1}{n}]} $$

Notice

$$ \int f_n = 1 \implies \lim \int f_n = 1 \implies \liminf \int f_n = 1$$

But, $\lim f_n = 0 \implies \liminf f_n = 0 \implies \int (\liminf f_n ) = 0 $

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There could be several reasons for which the equality does not hold:

  • if the measure space has infinite measure, mass can "escape", for example with $f_n:=\chi_{(n,n+1)}$ on the real line;
  • in the case of a finite measure space, there could be a huge diminution of the measure of the support of $f_n$ as $n$ goes to infinity;
  • oscillation of the function, for example $f_n(x):=\sin^2(n\pi x)$ on the unit interval.

It is worth mentioning that if $\sup_nf_n$ is integrable, then equality holds (dominated convergence theorem).

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