Probability measure: Difference between revisions

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[[File:Maxwell-Distr.png|thumb|300px|In some cases, [[statistical physics]] uses ''probability measures'', but not all [[measure theory|measures]] it uses are probability measures.<ref name=stern>''A course in mathematics for students of physics, Volume 2'' by Paul Bamberg, Shlomo Sternberg 1991 ISBN 0-521-40650-1 [http://books.google.com/books?id=eSmC4qQ0SCAC&pg=PA802 page 802]</ref><ref name= gut>''The concept of probability in statistical physics'' by Yair M. Guttmann 1999 ISBN 0-521-62128-3 [http://books.google.com/books?id=Q1AUhivGmyUC&pg=PA149 page 149]</ref>]]
{{Probability fundamentals}}
In mathematics, a '''probability measure''' is a [[real-valued function]] defined on a set of events in a [[probability space]] that satisfies [[Measure (mathematics)|measure]] properties such as ''countable additivity''.<ref>''An introduction to measure-theoretic probability'' by George G. Roussas 2004 ISBN 0-12-599022-7 [http://books.google.com/books?id=J8ZRgCNS-wcC&pg=PA47 page 47]</ref> The difference between a probability measure and the more general notion of measure (which includes concepts like [[area]] or [[volume]]) is that a probability measure must assign value 1 to the entire probability space.
 
Intuitively, the additivity property says that the probability assigned to the union of two disjoint events by the measure should be the sum of the probabilities of the events, e.g. the value assigned to "1 or 2" in a throw of a die should be the sum of the values assigned to "1" and "2".
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