# Definition:Simple Function

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## Definition

Let $\left({X, \Sigma}\right)$ be a measurable space.

A real-valued function $f: X \to \R$ is said to be a **simple function** if and only if it is a finite linear combination of characteristic functions:

- $\displaystyle f = \sum_{k \mathop = 1}^n a_k \chi_{S_k}$

where $a_1, a_2, \ldots, a_n$ are real numbers and each of the sets $S_k$ is $\Sigma$-measurable.

### Positive Simple Function

When all of the $a_i$ are positive, $f$ is also said to be **positive**.

## Also known as

When it is desirable to emphasize the $\sigma$-algebra $\Sigma$, one also speaks of **$\Sigma$-simple functions**.

## Also see

- Results about
**simple functions**can be found here.

## Sources

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