Lebesgue Integral

In mathematics, the integral of a non-negative function of one variable can be understood, in the simplest case, as the area between the graph of the function and the x-axis, and the Lebesgue integral, named after the French mathematician Henri Lebesgue, is one way of making that idea rigorous and extending it to a much wider class of functions. It is more general than the Riemann integral, which it has largely replaced in mathematical analysis since the first half of the twentieth century, because it can handle functions with discontinuities that are pathological from the Riemann integral's point of view and because it has better analytical properties, such as being able to exchange limits and integration under comparatively mild conditions. The Lebesgue integral also generalizes readily to more general measure spaces, including those used in probability theory, and the term Lebesgue integration can refer either to this general theory of integration with respect to an arbitrary measure or to the specific case of integrating a function on the real line with respect to Lebesgue measure. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/