A nonnegative measurable function 
is called Lebesgue integrable if its Lebesgue
integral 
is finite. An arbitrary measurable function is integrable if 
and 
are each Lebesgue integrable, where 
and 
denote the positive and
negative parts of 
, respectively.
The following equivalent characterization of Lebesgue integrable follows as a consequence of monotone convergence theorem. A
nonnegative measurable function 
is Lebesgue integrable iff there exists
a sequence of nonnegative simple
functions 
such that the following two conditions are satisfied:
1. 
.
