Definition:Simple Function

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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.

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.