# simple function

In measure theory, a 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 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.

 Title simple function Canonical name SimpleFunction Date of creation 2013-03-22 12:21:16 Last modified on 2013-03-22 12:21:16 Owner mps (409) Last modified by mps (409) Numerical id 9 Author mps (409) Entry type Definition Classification msc 03-00 Classification msc 26A09 Classification msc 26-00 Classification msc 28-00 Related topic CharacteristicFunction Related topic Integral2 Defines step function