Lebesgue Integral

measure-theory analysis

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 $f$, the Lebesgue integral is defined by approximating $f$ 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 $\mathbb{R}$ with the standard measure, the Lebesgue integral is written as

$$\int f(x)dx.$$

A function is Lebesgue integrable if its absolute value has finite integral:

$$\int |f|dx<\infty .$$