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[[File:MaxwellDistr.pngthumb300pxIn some cases, [[statistical physics]] uses ''probability measures'', but not all [[measure theorymeasures]] it uses are probability measures.<ref name=stern>''A course in mathematics for students of physics, Volume 2'' by Paul Bamberg, Shlomo Sternberg 1991
{{Probability fundamentals}}
In [[mathematics]], a '''probability measure''' is a [[realvalued 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 measuretheoretic probability'' by George G. Roussas 2004
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".
:<math>P(B \mid A) = \frac{P(A \cap B)}{P(A)}.</math>
satisfies the probability measure requirements so long as <math>P(A)</math> is not zero.<ref>''Probability, Random Processes, and Ergodic Properties'' by Robert M. Gray 2009
Probability measures are distinct from the more general notion of [[Fuzzy measure theoryfuzzy measures]] in which there is no requirement that the fuzzy values sum up to 1, and the additive property is replaced by an order relation based on [[set inclusion]].
==Example applications==
''Market measures'' which assign probabilities to [[financial market]] spaces based on actual market movements are examples of probability measures which are of interest in [[mathematical finance]], e.g. in the pricing of [[financial derivative]]s.<ref>''Quantitative methods in derivatives pricing'' by Domingo Tavella 2002
Not all measures that intuitively represent chance or likelihood are probability measures. For instance, although the fundamental concept of a system in [[statistical mechanics]] is a measure space, such measures are not always probability measures.<ref name=stern/> In general, in statistical physics, if we consider sentences of the form "the probability of a system S assuming state A is p" the geometry of the system does not always lead to the definition of a probability measure [[congruence relationunder congruence]], although it may do so in the case of systems with just one degree of freedom.<ref name=gut/>
Probability measures are also used in [[mathematical biology]].<ref>''Mathematical Methods in Biology'' by J. David Logan, William R. Wolesensky 2009
==See also==
==Further reading==
*''Probability and Measure'' by [[Patrick Billingsley]], 1995 John Wiley
*''Probability & Measure Theory'' by Robert B. Ash, Catherine A. DoléansDade 1999 Academic Press
[[Category:Measures (measure theory)]]
