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 $f$:

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

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.

This page has been identified as a candidate for refactoring of basic complexity. In particular: Sort out categories and cross-links throughout this area Until this has been finished, please leave {{Refactor}} in the code. New contributors: Refactoring is a task which is expected to be undertaken by experienced editors only. Because of the underlying complexity of the work needed, it is recommended that you do not embark on a refactoring task until you have become familiar with the structural nature of pages of $\mathsf{Pr} \infty \mathsf{fWiki}$. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Refactor}} from the code.

• 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