Step function: Difference between revisions

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* 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>

*A [[discrete random variable]] is defined as a [[random variable]] whose [[cumulative distribution function]] is piecewise constant.<ref name=":0">{{Cite book|url=https://www.worldcat.org/oclc/51441829|title=Introduction to Probability|last=Bertsekas|first=Dimitri P.|date=2002|publisher=Athena Scientific|others=Tsitsiklis, John N., Τσιτσικλής, Γιάννης Ν.|year=|isbn=188652940X|___location=Belmont, Mass.|pages=|oclc=51441829}}</ref>

==See also==