Real Integral Monotone Convergence
Why
An integral is a limit. When can we exchange this limit with another? We being a development culminating in a sufficient condition for exchange.
Result
The integral of the almost everywhere limit of an almost-everywhere nondecreasing sequence of measurable, nonnegative, extended-real-valued functions is the limit of the sequence of integrals of the functions.
Suppose $(X, \mathcal{A} , \mu )$ is a measure
space, $\seqt{f}: X \to \eri$ is a
$\mathcal{A} $-measurable function for every
natural number $n$ and $f: X \to \eri$ is a
$\mathcal{A} $-measurable function.
If
\[ f_n(x) \leq f_{n+1}(x) \quad \text{ and } \quad f(x) = \lim_n \seqt{f}(x) \]
hold for all natural $n$ and almost every $x$ in $X$, then\[ \int f d\mu = \lim_n \int f_n d\mu . \]
This result is often called the monotone convergence theorem.