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Fatou's lemma states that if $f_1,f_2,\ldots$ are nonnegative measurable functions, and $f=\liminf f_i$, then $$\int_E fd\mu\leq \liminf\int_E f_id\mu$$ The two examples for strict inequality given in the link are where $f_n$ takes on a nonzero value on an arbitrarily large domain, and where $f_n$ gets arbitrarily large.

If we restrict the domain to $[0,1]$ and require $0\leq f_i\leq 1$, does equality always hold?

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    $\begingroup$ No. $f_i = \chi_{[0,1/2]}$ if $i$ is even, and $f_i = \chi_{[1/2,1]}$ if $i$ is odd. $\endgroup$ – Daniel Fischer Oct 22 '13 at 14:19
  • $\begingroup$ @DanielFischer IMO that's an answer. $\endgroup$ – Jonathan Y. Oct 22 '13 at 14:33
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No, even with bounded functions on a space of finite measure we can find examples for the strict inequality, if the space has two disjoint sets $A,B$ with positive measure. Then one can simply oscillate

$$f_i = \begin{cases}\chi_A &, i \equiv 0 \pmod{2}\\ \chi_B &, i \equiv 1 \pmod{2} \end{cases}$$

and one has $f = \liminf f_i = 0$, but $\liminf \int f_i = \min \{\mu(A),\,\mu(B)\} > 0$.

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