Definition
Lebesgue Integral
The Lebesgue integral is a way of integrating a measurable function with respect to a measure. It generalises the Riemann integral and is better suited to functions with complicated sets of discontinuities.
For a non-negative measurable function
, the Lebesgue integral is defined by approximating with simpler measurable functions and taking the supremum of their integrals. For a general measurable function, one integrates its positive and negative parts separately. In the most common case, on
with the standard measure, the Lebesgue integral is written as A function is Lebesgue integrable if its absolute value has finite integral: