A Lebesgue integrable function on a measure space 
with a positive measure 
is a measurable function ![f:X->[-infty,infty]](/images/equations/LebesgueIntegrable/Inline3.svg)
or 
satisfying 
(Hunter 2011, pp. 42 and 50). For
a nonnegative measurable
function, this means that its Lebesgue integral
is finite. For ![f:X->[-infty,infty]](/images/equations/LebesgueIntegrable/Inline6.svg)
, this is
equivalent to both 
and 
being Lebesgue integrable, where 
and 
are its positive part and
negative part, respectively.
The following equivalent characterization follows from the monotone convergence theorem. A nonnegative measurable
function 
is Lebesgue integrable iff there exists a sequence
of nonnegative measurable simple functions 
satisfying:
1. 
.
Indeed, choose increasing nonnegative measurable simple functions 
converging to 
, put 
, and take 
. Their sum
is 
, and the monotone
convergence theorem gives 
. Conversely,
applying that theorem to the partial sums 
proves that the two conditions imply Lebesgue
integrability (Hunter 2011, pp. 38 and 41-42).
