The Lebesgue integral is defined with respect to a positive measure 
on a measure space 
. For a nonnegative measurable function 
, its integral is the supremum
of the integrals of all nonnegative measurable simple functions 
satisfying 
. It may equal 
. Equivalently, it is obtained by approximating
from below by an increasing sequence
of such simple functions.
For example, with Lebesgue measure, the integral of a nonnegative measurable simple function is a Lebesgue
sum 
, where the set 
consists of the points at which that
function has value 
. These sets need not be
intervals. The definition extends to signed measurable
functions by taking the difference of the integrals of their positive
parts and negative parts, provided at least
one is finite (Hunter 2011, pp. 39-42). This type of integral covers a wider
class of functions than does the Riemann
integral.
A measurable function 
with real part 
and imaginary part 
is Lebesgue
integrable iff 
and 
are both Lebesgue integrable.
In that case, 
, and these integrals are
finite (Hunter 2011, p. 50).
The Lebesgue integral of a function 
over a measure space 
is written
 |
or sometimes
 |
to emphasize that the integral is taken with respect to the measure 
.
