simple function

In measure theory, a simple functionMathworldPlanetmathPlanetmath is a function that is a finite linear combinationMathworldPlanetmath


of characteristic functionsMathworldPlanetmathPlanetmathPlanetmath, where the ck are real coefficients and every Ak is a measurable setMathworldPlanetmath with respect to a fixed measure spaceMathworldPlanetmath.

If the measure space is and each Ak is an interval, then the function is called a step function. Thus, every step function is a simple function.

Simple functions are used in analysisMathworldPlanetmath to interpolate between characteristic functions and measurable functionsMathworldPlanetmath. In other words, characteristic functions are easy to integrate:


while simple functions are not much harder to integrate:


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