Simple function

In mathematics, particularly in calculus, a simple function is a function which is measurable and takes only finitely many values ​​. This is the range of values ​​, or more generally a Banach space. Simple functions play a central role in the theory of integration.

A simple function is also referred to as an elementary function incorrectly as step function.

Definition

Let be a measurable space and a ( real or complex ) Banach space. A function is called a simple function if the following conditions are met:

• Only finally takes on many values
• Is measurable, that is valid for all.

Is even defined on a measure space, we sometimes require additionally that

Is finite.

This is equivalent to the function of a representation of the form

Possesses. It is referred to, and the characteristic function of the measurable quantity. This representation is called canonical.

Properties

Sums, differences and products of simple functions are again easy, just scalar multiples. Thus, the space of simple functions forms a commutative algebra over or.

Use

Simple functions play a central role in the definition of the Lebesgue integral and the Bochner integral. Here, the integral is first for positive simple functions

Defined and then transferred by approximation to additional functions.

Confusion with step functions

Simple functions to be confused with step functions often used to define the Riemann integral. Both functions take only finitely many function values ​​. A step function is, however, only a finite number of intervals on which it has constant function values ​​. A simple function, however, can, for example, on any number of intervals always have two function values ​​alternately, making it no longer a step function. In particular, the indicator function of the rational numbers ( Dirichlet function) is a simple function, although it is not riemann integrable.