# Fatou's lemma on bounded domain and bounded range

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?

• No. $f_i = \chi_{[0,1/2]}$ if $i$ is even, and $f_i = \chi_{[1/2,1]}$ if $i$ is odd. – Daniel Fischer Oct 22 '13 at 14:19
• @DanielFischer IMO that's an answer. – Jonathan Y. Oct 22 '13 at 14:33

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 $\liminf \int f_i = \min \{\mu(A),\,\mu(B)\} > 0$.