PlanetMath: simple function

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simple function

(Definition)

In measure theory, a simple function is a function that is a finite linear combination

$\displaystyle h = \sum_{k=1}^n c_k \chi_{A_k}$

of characteristic functions, where the

are real coefficients and every

is a measurable set with respect to a fixed measure space.

If the measure space is $\mathbb{R}$ and each is an interval, then the function is called a 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:

$\displaystyle \int_E \chi_{A}\,dx = \vert A\vert,$

while simple functions are not much harder to integrate:

$\displaystyle \int_E \sum_{k=1}^n c_k \chi_{A_k}\,dx = \sum_{k=1}^n c_k \vert A_k\vert.$

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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"simple function" is owned by mps. [ full author list (3) | owner history (1) ]

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Also defines:

step function

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Cross-references: Lebesgue integral over a subset of the measure space, measurable functions, interval, measure space, fixed, measurable set, coefficients, real, characteristic functions, linear combination, finite, function, theory, measure
There are 8 references to this entry.

This is version 6 of simple function, born on 2002-02-16, modified 2007-06-29.
Object id is 2022, canonical name is SimpleFunction.
Accessed 4163 times total.

Classification:

AMS MSC:	03-00 (Mathematical logic and foundations :: General reference works )
	26-00 (Real functions :: General reference works )
	26A09 (Real functions :: Functions of one variable :: Elementary functions)
	28-00 (Measure and integration :: General reference works )

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