# Definition:Lebesgue Integral

## Definition

Let $\lambda^n$ be a Lebesgue measure on $\R^n$.

Let $f: \R^n \to \overline{\R}$ be a Lebesgue integrable function.

Then the $\lambda^n$-integral of $f$:

$\displaystyle \int f \, \mathrm d \lambda^n$

is called the Lebesgue integral of $f$.

## Also known as

Historically, the notations $\displaystyle \int f \left({x}\right) \, \mathrm d x$ and $\displaystyle \int f \, \mathrm d x$ are used in place of the formally correct $\displaystyle \int f \, \mathrm d \lambda^n$.

## Source of Name

This entry was named for Henri Léon Lebesgue.

## Historical Note

The concept of a Lebesgue integral was inspired by Bernhard Riemann's own definition of an integral in his paper Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe of $1854$, on the subject of Fourier series.