Theorem (monotone convergence theorem):
Let
be a measure space, and let
be an ascending (that is,
pointwise) sequence of non-negative functions, that converges pointwise to a function
. Then
.
Theorem (Fatou's lemma):
Let
be a measure space, and let
be a sequence of non-negative functions. Then
.
Proof: Note that, upon defining
,
that the sequence of functions
is strictly ascending and converges pointwise to
as
. Hence, the monotone convergence theorem is applicable and we obtain
.
Now for each
, we have
,
and if we take the lim inf,
. ![{\displaystyle \Box }](https://wikimedia.org/api/rest_v1/media/math/render/svg/029b77f09ebeaf7528fc831fe57848be51f2240b)