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 $f$:

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

is called the Lebesgue integral of $f$.

Also known as

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

Also see

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

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

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