Lebesgue integration

A Lebesgue integral has the same value as the Riemann integral, when the latter exists. However, Lebesgue integrals exist for many more functions. The Lebesgue integral for positive can be defined as the Riemann integral of the function x |-> x*m({p: f(p)>=x}). It uses the notion of measure and any measurable positive function has a Lebesgue integral. Any finite function on a finite interval has a Lebesgue integral, defined as the Lebesgue integral of its positive part minus the Lebesgue integral as its negative part.

See null set, [Henri Leon Lebesgue]?,integration,measure