Definition:Measurable Function
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Definition 1
Let $\left({X, \Sigma}\right)$ be a measurable space.
Let $E \in \Sigma$.
Then a function $f: E \to \R$ is said to be $\Sigma$-measurable on $E$ iff:
- $\forall \alpha \in \R: \left\{{x \in E : f \left({x}\right) \le \alpha}\right\} \in \Sigma$
See the theorem on measurable images for equivalences of this definition.
Definition 2
Real-Valued Function
Let $\left({X, \Sigma}\right)$ be a measurable space.
Let $\mathcal B$ be the Borel $\sigma$-algebra on $\R$.
A real-valued function $f: X \to \R$ is said to be ($\Sigma$-)measurable iff $f$ is $\Sigma \, / \, \mathcal B$-measurable.
Extended Real-Valued Function
Let $\left({X, \Sigma}\right)$ be a measurable space.
Let $\overline{\mathcal B}$ be the Borel $\sigma$-algebra on the extended real number space.
An extended real-valued function $f: X \to \R$ is said to be ($\Sigma$-)measurable iff $f$ is $\Sigma \, / \, \overline{\mathcal B}$-measurable.
Def 1 corresponds to def 2 for real-valued fns; this needs to be covered
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Positive Measurable Function
A measurable function $f$ is said to be a positive measurable function iff it also satisfies:
- $f \ge 0$
where $\ge$ denotes pointwise inequality.
Define the notion of $\Sigma$-measurable function on a subset
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Also see
- Measurable Mapping, a more general notion of which this is a specialization.
- Space of Measurable Functions, naming collections of $\Sigma$-measurable functions conveniently
Sources
- 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $\S 8$
