Definition:Space of Simple Functions
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Definition
Let $\struct {X, \Sigma}$ be a measurable space.
Then the space of simple functions on $\struct {X, \Sigma}$, denoted $\map \EE \Sigma$, is the collection of all simple functions $f: X \to \R$:
- $\map \EE \Sigma := \set {f: X \to \R: \text{$f$ is a simple function} }$
Space of Positive Simple Functions
The space of positive simple functions on $\struct {X, \Sigma}$, denoted $\map {\EE^+} \Sigma$, is the subset of positive simple functions in $\map \EE \Sigma$:
- $\map {\EE^+} \Sigma := \set {f: X \to \R: \text {$f$ is a positive simple function} }$
Also known as
Often, one simply speaks about the space of (positive) simple functions when the measurable space $\struct {X, \Sigma}$ is understood.
It is also common to write $\EE$ (resp. $\EE^+$) instead of $\map \EE \Sigma$ (resp. $\map {\EE^+} \Sigma$) when $\Sigma$ is clear from the context.
Also see
Sources
- 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $8.6$