View source for Probability measure

{{Use American English|date = March 2019}}
{{Short description|Measure of total value one, generalizing probability distributions}}
[[File:Maxwell-Distr.png|thumb|300px|In many 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}} [https://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}} [https://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}} [https://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".

Probability measures have applications in diverse fields, from physics to finance and biology.

==Definition==
[[File:Probability-measure.svg|thumb|300px|A ''probability measure'' mapping the probability space for 3 events to the [[unit interval]].]]
The requirements for a function {{math|''&mu;''}} to be a probability measure on a [[probability space]] are that:

:*{{math|''&mu;''}} must return results in the [[unit interval]] [0, 1], returning 0 for the empty set and 1 for the entire space.

:*{{math|''&mu;''}} must satisfy the ''countable additivity'' property that for all [[countable]] collections <math>\{E_i\}</math> of pairwise [[disjoint sets]]: 
::<math> \mu\Bigl(\bigcup_{i \in I} E_i\Bigr) = \sum_{i \in I} \mu(E_i).</math>

For example, given three elements 1, 2 and 3 with probabilities 1/4, 1/4 and 1/2, the value assigned to {1, 3} is 1/4 + 1/2 = 3/4, as in the diagram on the right.

The [[conditional probability]] based on the intersection of events defined as:

:<math>\mu (B \mid A) = \frac{\mu(A \cap B)}{\mu(A)}.</math>

satisfies the probability measure requirements so long as <math>\mu(A)</math> is not zero.<ref>''Probability, Random Processes, and Ergodic Properties'' by Robert M. Gray 2009 {{isbn|1-4419-1089-1}} [https://books.google.com/books?id=x-VbL8mZWl8C&pg=PA163 page 163]</ref>

Probability measures are distinct from the more general notion of [[Fuzzy measure theory|fuzzy 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 {{isbn|0-471-39447-5}} [https://books.google.com/books?id=dHIMulKy8dYC&pg=PA11 page 11]</ref> For instance, a [[risk-neutral measure]] is a probability measure which assumes that the current value of assets is the [[expected value]] of the future payoff taken with respect to that same risk neutral measure (i.e. calculated using the corresponding risk neutral density function), and  [[discounted]] at the [[risk-free rate]]. If there is a unique probability measure that must be used to price assets in a market, then the market is called a [[complete market]].<ref>''Irreversible decisions under uncertainty'' by Svetlana I. Boyarchenko, Serge Levendorskiĭ 2007 {{isbn|3-540-73745-6}} [https://books.google.com/books?id=lpsrP5mQG_QC&pg=PA11 page 11]</ref>

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 relation|under 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 {{isbn|0-470-52587-8}} [https://books.google.com/books?id=6GGyquH8kLcC&pg=PA195 page 195]</ref> For instance, in comparative [[sequence analysis]] a probability measure may be defined for the likelihood that a variant may be permissible for an [[amino acid]] in a sequence.<ref>''Discovering biomolecular mechanisms with computational biology'' by Frank Eisenhaber 2006 {{isbn|0-387-34527-2}} [https://books.google.com/books?id=Pygg7cIZTwIC&pg=PA127 page 127]</ref>

==See also==
*[[Borel measure]]
*[[Fuzzy measure]]
*[[Haar measure]]
*[[Martingale measure]]
*[[Lebesgue measure]]

==References==
{{Reflist}}

==Further reading==
*{{cite book |title=Probability and Measure |first=Patrick |last=Billingsley |authorlink=Patrick Billingsley |year=1995 |publisher=John Wiley |isbn=0-471-00710-2 }}
*{{cite book |title=Probability & Measure Theory |first=Robert B. |last=Ash |first2=Catherine A. |last2=Doléans-Dade |year=1999 |publisher=Academic Press |isbn= 0-12-065202-1}}

==External links==
*{{Commonscat-inline}}

[[Category:Measures (measure theory)]]
[[Category:Experiment (probability theory)]]

[[pl:Miara probabilistyczna]]