Lebesgue's dominated convergence theorem states that if 
is a sequence of measurable
functions 
or 
on a measure space
with a positive measure 
, 
is measurable, 
converges pointwise to 
almost everywhere as
,
and 
almost everywhere
for all 
, where 
is a nonnegative Lebesgue
integrable function, then 
is Lebesgue integrable,
and
 |
(Hunter 2011, pp. 49-50).
If the measure is not a complete measure, an almost everywhere limit of measurable functions need not be measurable.
