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 $\displaystyle \int f \rd \lambda^n$.
- Definition:Integral of Integrable Function, of which the Lebesgue integral is an instance.
Source of Name
This entry was named for Henri Léon Lebesgue.
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.