Real Integral Limit Inferior Bound
Result
The integral of the limit inferior of a sequence of measurable, nonnegative, extended-real-valued functions is no larger than the limit inferior of the sequence of integrals.
Let $(X, \mathcal{A} , \mu )$ be a measure
space, and let $\seqt{f}: \to \nneri$ a
$\mathcal{A} $-measurable function for every
natural number $n$.
We want to show that if
\[
\int \liminf_n f_n d\mu \leq \liminf_n \int f_n d\mu .
\]