simple functionSimpleFunction2002-02-16 20:52:122007-06-29 00:37:22Definitionstep function\usepackage{amssymb}
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\newcommand{\braket}[2]{\langle #1 \ket{#2}}In measure theory, a \emph{simple function} is a function that is a
finite linear combination
\[
h = \sum_{k=1}^n c_k \chi_{A_k}
\]
of characteristic functions, where the $c_k$ are real coefficients and
every $A_k$ is a measurable set with respect to a fixed measure space.
If the measure space is $\mathbb{R}$ and each $A_k$ is an interval,
then the function is called a \emph{step function}. Thus, every step
function is a simple function.
Simple functions are used in analysis to interpolate between
characteristic functions and measurable functions. In other words,
characteristic functions are easy to integrate:
\[
\int_E \chi_{A}\,dx = |A|,
\]
while simple functions are not much harder to integrate:
\[
\int_E \sum_{k=1}^n c_k \chi_{A_k}\,dx = \sum_{k=1}^n c_k |A_k|.
\]
To integrate a measurable function, one approximates it from below by
simple functions. Thus, simple functions can be used to define the
Lebesgue integral over a subset of the measure space.
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