Simple function

In mathematics , especially in analysis , a simple function is a function that can be measured and only takes on a finite number of values. The range of values or, more generally, is a Banach space . Simple functions play a central role in integration theory . ${\displaystyle \mathbb {R} }$

A simple function is also called an elementary function or a step function .

definition

Let be a measurement space and a (real or complex) Banach space . A function is called a simple function if the following conditions are met: ${\displaystyle (X,\Sigma )}$ ${\displaystyle V}$ ${\displaystyle u\colon X\to V}$

• ${\displaystyle u}$takes only finitely many values of ${\displaystyle \{v_{1},\ldots ,v_{n}\}}$
• ${\displaystyle u}$is measurable , i.e. H. for all true . ${\displaystyle v\in V}$ ${\displaystyle u^{-1}(\{v\})\in \Sigma }$

If it is even defined on a dimension space , one also sometimes demands that ${\displaystyle u\colon X\to V}$ ${\displaystyle (X,\Sigma ,\mu )}$

• ${\displaystyle \mu (u^{-1}(V\setminus \{0\}))}$

is finite.

It is equivalent to this that the function is a representation of the form ${\displaystyle u}$

${\displaystyle u(x)=\sum _{i=1}^{n}v_{i}\cdot \chi _{E_{i}}(x)}$

owns. Here is and denotes the characteristic function of the measurable amount . This representation is called canonical . ${\displaystyle v_{i}\in V}$ ${\displaystyle \chi _{E_{i}}}$ ${\displaystyle E_{i}=u^{-1}(\{v_{i}\})\in \Sigma }$

properties

Sums, differences and products of simple functions are simple again, as are scalar multiples. Thus the space of simple functions forms a commutative algebra over or . ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {C} }$

use

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

${\displaystyle \int _{\Omega }u\,{\rm {d}}\mu :=\sum _{i=1}^{m}v_{i}\mu (E_{i})}$

and then transferred to other functions by approximation. Here is one of the finitely many values ​​of the simple function . is the set of values ​​for which is equal . ${\displaystyle v_{i}}$ ${\displaystyle u}$ ${\displaystyle E_{i}=u^{-1}(\{v_{i}\})}$ ${\displaystyle u}$ ${\displaystyle v_{i}}$

Differentiation from staircase functions

Often simple functions are confused with step functions that are used to define the Riemann integral . Both functions only take on a finite number of function values. However, a step function only consists of a finite number of intervals on which it has constant function values. A simple function, on the other hand, can, for example, assume two function values ​​alternately at any number of intervals and is therefore no longer a step function. In particular, the indicator function of the rational numbers ( Dirichlet function ) is a simple function, although it cannot be Riemann-integrable. ${\displaystyle \chi _{\mathbb {Q} }}$

literature

• Richard M. Dudley: Real Analysis and Probability (= Cambridge Studies in Advanced Mathematics. Vol. 74). Cambridge University Press, Cambridge et al. 2002, ISBN 0-521-80972-X , pp. 114-7.
• David Meintrup, Stefan Schäffler: Stochastics. Theory and applications. Berlin, Heidelberg et al. 2005, ISBN 3-540-21676-6 .

Individual evidence

1. ↑ Achim Klenke: Probability Theory . 3. Edition. Springer-Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-36017-6 , p. 39 , doi : 10.1007 / 978-3-642-36018-3 .
2. ^ Hans Wilhelm Alt : Linear functional analysis . 6th edition. Springer-Verlag, Berlin Heidelberg 2012, ISBN 978-3-642-22260-3 , p. 77 , doi : 10.1007 / 978-3-642-22261-0 .
3. ↑ Dirk Werner : Functional Analysis . 7th, corrected and enlarged edition. Springer-Verlag, Heidelberg Dordrecht London New York 2011, ISBN 978-3-642-21016-7 , pp.  32 , doi : 10.1007 / 978-3-642-21017-4 .
4. ^ Herbert Amann, Joachim Escher : Analysis. Volume 3. Birkhäuser, Basel et al. 2001, ISBN 3-7643-6613-3 , p. 65.