Definition:Integral of Measure-Integrable Function

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Definition

Let $\struct {X, \Sigma, \mu}$ be a measure space.

Let $f: X \to \overline \R$, $f \in \map {\LL^1} \mu$ be a $\mu$-integrable function.


Then the $\mu$-integral of $f$ is defined by:

$\ds \int f \rd \mu := \int f^+ \rd \mu - \int f^- \rd \mu$

where $f^+$ and $f^-$ are the positive and negative parts of $f$, respectively.


Complex Function

Let $\struct {X, \Sigma, \mu}$ be a measure space.

Let $\struct {\C, \map \BB \C}$ be the complex numbers made into a measurable space with its Borel $\sigma$-algebra.

Let $f : X \to \C$ be a $\mu$-integrable function.

Let $\map \Re f : X \to \R$ and $\map \Im f : X \to \R$ be the real part and imaginary part of $f$ respectively.


We define the integral of $f$ by:

$\ds \int f \rd \mu = \int \map \Re f \rd \mu + i \int \map \Im f \rd \mu$


Also known as

The $\mu$-integral is also sometimes called the (abstract) Lebesgue integral.

The name Lebesgue integral is a tribute to Henri Léon Lebesgue, one of the founders of measure theory.

Use of this name is discouraged as there is possible confusion with the notion of Lebesgue integral (which is an instance of the concept here defined).


Also see


Sources

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