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 $ℝ$ 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:

$${\int }_E{\chi }_A𝑑x=|A|,$$

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