Definition:Integrable Function/Measure Space
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Definition
Let $\struct {X, \Sigma, \mu}$ be a measure space.
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.
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
- Definition:Integral of Integrable Function, justifying the name integrable function
- Definition:Space of Integrable Functions
- Characterization of Integrable Functions, demonstrating other ways to verify $\mu$-integrability.
Sources
- 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $10.1$