# Definition:Simple Function

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## Definition

Let $\struct {X, \Sigma}$ 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:

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

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**.

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### Banach Space

Let $\GF \in \set {\R, \C}$.

Let $I$ be a real interval.

Let $X$ be a Banach space over $\GF$.

Let $f : I \to X$ be a function.

We say that $f$ is **simple** if and only if there exists:

- Lebesgue measurable subsets $\Omega_1, \ldots, \Omega_r$ of $I$ with finite Lebesgue measure
- $x_1, \ldots, x_r \in X$

such that:

- $\ds \map f t = \sum_{r \mathop = 1}^n x_r \map {\chi_{\Omega_r} } t$

for each $t \in I$.

## 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

- 1991: David Williams:
*Probability with Martingales*... (previous) ... (next): $5.1$: Integrals of non-negative simple functions, $SF^+$ - 2005: René L. Schilling:
*Measures, Integrals and Martingales*... (previous) ... (next): $8.6$