Probability measure: Difference between revisions

explain distinction between prob measure and general measure, expand a bit on additivity
(improve refs)
(explain distinction between prob measure and general measure, expand a bit on additivity)
[[File:Atomisation-Cl-3D-vdW.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 0521406501 [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 0521621283 [http://books.google.com/books?id=Q1AUhivGmyUC&pg=PA149 page 149]</ref>]]
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 0125990227 [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 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 "Heads or Tails" in a coin toss should be the sum of the values assigned to Heads and Tails.
 
Probability measures have applications in diverse fields, from physics to finance and biology.