thus begins with a measure, u. In the simplest case, the Lebesgue measure of an interval is its width, , so that the Lebesgue integral agrees with the (proper) Riemann integral when both exist. In more complicated cases, the sets being measured can be highly fragmented, with no continuity and no resemblance to intervals.

Using the "partitioning the range of " philosophy, the integral of a non-negative function should be the sum over of the