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 $\displaystyle \int f \, \mathrm d \lambda^n$.
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): $\S 10$
