Definition:Integrable Function/Measure Space

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Definition

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

Let $\MM_{\overline \R}$ denote the space of $\Sigma$-measurable, extended real-valued functions .

Let $f \in \MM_{\overline \R}, f: X \to \overline \R$ be a measurable function.


Then $f$ is said to be $\mu$-integrable if and only if:

$\ds \int f^+ \rd \mu < +\infty$

and

$\ds \int f^- \rd \mu < +\infty$

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


The integral signs denote $\mu$-integration of positive measurable functions.


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 $\Sigma/\map \BB C$-measurable 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 say that $f$ is $\mu$-integrable if and only if $\map \Re f$ and $\map \Im f$ are integrable.


Also known as

When no ambiguity arises, one may also simply speak of integrable functions.

To emphasize $X$ or $\Sigma$, also $X$-integrable function and $\Sigma$-integrable function are encountered.

Any possible ambiguity may be suppressed by the phrasing $\struct {X, \Sigma, \mu}$-integrable functions, but this is usually too cumbersome.


Also see

  • Results about integrable functions can be found here.


Sources

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