World Library  
Flag as Inappropriate
Email this Article

Counting measure

Article Id: WHEBN0000191786
Reproduction Date:

Title: Counting measure  
Author: World Heritage Encyclopedia
Language: English
Subject: Measure (mathematics), Point process notation, Σ-finite measure, Lp space, Measures (measure theory)
Collection: Measures (Measure Theory)
Publisher: World Heritage Encyclopedia
Publication
Date:
 

Counting measure

In mathematics, the counting measure is an intuitive way to put a measure on any set: the "size" of a subset is taken to be the number of elements in the subset, if the subset has finitely many elements, and if the subset is infinite.[1]

The counting measure can be defined on any measurable set, but is mostly used on countable sets.[1]

In formal notation, we can make any set X into a measurable space by taking the sigma-algebra \Sigma of measurable subsets to consist of all subsets of X. Then the counting measure \mu on this measurable space (X,\Sigma) is the positive measure \Sigma\rightarrow[0,+\infty] defined by

\mu(A)=\begin{cases} \vert A \vert & \text{if } A \text{ is finite}\\ +\infty & \text{if } A \text{ is infinite} \end{cases}

for all A\in\Sigma, where \vert A\vert denotes the cardinality of the set A.[2]

The counting measure on (X,\Sigma) is σ-finite if and only if the space X is countable.[3]

Discussion

The counting measure is a special case of a more general construct. With the notation as above, any function f \colon X \to [0, \infty) defines a measure \mu on (X, \Sigma) via

\mu(A):=\sum_{a \in A} f(a)\ \forall A\subseteq X,

where the possibly uncountable sum of real numbers is defined to be the sup of the sums over all finite subsets, i.e.,

\sum_{y \in Y \subseteq \mathbb R} y := \sup_{F \subseteq Y, |F| < \infty} \left\{ \sum_{y \in F} y \right\}.

Taking f(x)=1 for all x in X produces the counting measure.

Notes

  1. ^ a b Counting Measure at PlanetMath.org.
  2. ^ Schilling (2005), p.27
  3. ^ Hansen (2009) p.47

References

  • Schilling, René L. (2005)."Measures, Integral and Martingales". Cambridge University Press.
  • Hansen, Ernst (2009)."Measure theory, Fourth Edition". Department of Mathematical Science, University of Copenhagen.
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
 
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
 
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.
 


Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.