# 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): $8.6$