Definition:Lebesgue Integral

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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

  • 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.