# Definition:Lebesgue Integral

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## 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 $f$:

- $\ds \int f \rd \lambda^n$

is called the **Lebesgue integral of $f$**.

## Also known as

Historically, the notations $\ds \int \map f x \rd x$ and $\ds \int f \rd x$ are used in place of the formally correct $\ds \int f \rd \lambda^n$.

## Also see

- Definition:Integral of Integrable Function, of which the
**Lebesgue integral**is an instance.

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- Results about
**Lebesgue integrals**can be found**here**.

## 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.

## Sources

- 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**Lebesgue integral** - 2005: René L. Schilling:
*Measures, Integrals and Martingales*... (previous) ... (next): $\S 10$ - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**Lebesgue integral**