Lebesgue-Integrable Complex-Valued Function

measure-theory analysis

Definition

Lebesgue-Integrable Function

A Lebesgue-integrable function is a complex-valued function whose absolute value has a finite Lebesgue integral over its domain.

For a function $f:[a,b]\to \mathbb{C}$, this means

$${\int }_a^b|f(x)|dx<\infty .$$

Such functions are the Lebesgue-integrable functions on $[a,b]$.

If $f$ is real-valued, the condition becomes

$${\int }_a^b|f(x)|dx<\infty .$$