Real Integral Dominated Convergence
Why
An integral is a limit. When can we exchange this limit with another? We give a first result in the search for sufficient conditions to do so.
Result
When context is clear, we refer to the following proposition as the dominated convergence theorem.
The integral of the almost everywhere limit of a sequence of measurable, extended-real-valued, almost-everywhere bounded functions is the limit of the sequence of integrals of the functions.
Let $(X, \mathcal{A} , \mu )$
be a measure space.
Let $f: X \to \eri$ be a
$\mathcal{A} $-measurable function.
Let $\seqt{f}: \to \eri$
a $\mathcal{A} $-measurable function
for every natural number $n$ so that
$\seq{f}$ converges almost everywhere
to $f$.
Let $g: X \to \nneri$ be an
integrable function which
dominates $f_n$ almost everywhere
for each $n$.
We want to show that:
\[
\int f d\mu = \lim_{n} \int f_n d\mu .
\]