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.