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

• Simple Function
• Space of Simple Functions forms Algebra

Sources

• 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $8.6$