The Lebesgue integral is defined in terms of upper and lower bounds using the Lebesgue measure of a set.
It uses a Lebesgue sum 
where 
is the value of the function in subinterval 
, and 
is the Lebesgue measure
of the set 
of points for which values are approximately 
. This type of integral covers a wider class of functions
than does the Riemann integral.
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 
.
