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): $8.6$