Step function: Difference between revisions

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==Definition and first consequences==

A function <math>f\colon \mathbb{R} \rightarrow \mathbb{R}</math> is called a '''step function''' if it can be written as {{Citation needed|date=September 2009}}

:<math>f(x) = \sum\limits_{i=0}^n \alpha_i \chi_{A_i}(x)\ \forall </math>, for all real numbers <math>x</math>

where <math>n\ge 0</math> and <math>\alpha_i</math> are real numbers, <math>A_i</math> are intervals, and <math>\chi_A</math> is the [[indicator function]] of <math>A\text{:}</math>

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* The sum and product of two step functions is again a step function. The product of a step function with a number is also a step function. As such, the step functions form an [[algebra over a field|algebra]] over the real numbers.

* A step function takes only a finite number of values. If the intervals <math>A_i,</math> for <math>i=0, 1, \dots, n</math> in the above definition of the step function are disjoint and their union is the real line, then <math>f(x)=\alpha_i\</math> \forall\for all <math>x\in A_i.</math>

* The [[definite integral]] of a step function is a [[piecewise linear function]].

* The [[Lebesgue integral]] of a step function <math>\textstyle f = \sum\limits_{i=0}^n \alpha_i \chi_{A_i}</math> is <math>\textstyle \int f\,dx = \sum\limits_{i=0}^n \alpha_i \ell(A_i),\,</math> where <math>\textstyle\ell(A)</math> is the length of the interval <math>A,</math> and it is assumed here that all intervals <math>A_i</math> have finite length. In fact, this equality (viewed as a definition) can be the first step in constructing the Lebesgue integral.<ref>{{Cite book | author=Weir, Alan J | authorlink= | coauthors= | title=Lebesgue integration and measure | date= | publisher=Cambridge University Press, 1973 | ___location= | isbn=0-521-09751-7 |chapter= 3}}</ref>