A Lebesgue sum is the integral of a simple function used to approximate a measurable function
in the construction of the Lebesgue integral.
For a nonnegative measurable simple function 
, it is
 |
where 
is a positive measure on a measure
space 
,
the 
are pairwise disjoint
sets that are measurable, 
, and 
is the characteristic
function of 
.
The convention 
is used. For example, the 
may be the sets on which 
takes its distinct values (Hunter 2011, p. 39).
Taking an increasing sequence of these simple functions that converges to a nonnegative measurable function 
gives 
by the monotone
convergence theorem. With Lebesgue measure
on 
, the sets 
need not be intervals;
the approximation groups values of the function rather
than requiring a partition of the domain
into intervals (Hunter 2011, pp. 38-41).
