Editing Probability measure

{{Short description|Measure of total value one, generalizing probability distributions}}
{{Use American English|date = March 2019}}
{{Probability fundamentals}}
In [[mathematics]], a '''probability measure''' is a [[real-valued function]] defined on a set of events in a [[σ-algebra]] 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 space.

Intuitively, the additivity property says that the probability assigned to the union of two disjoint (mutually exclusive) events by the measure should be the sum of the probabilities of the events; for example, the value assigned to the outcome "1 or 2" in a throw of a dice should be the sum of the values assigned to the outcomes "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 σ-algebra for  <math>2^3</math> events to the [[unit interval]].]]
The requirements for a [[set function]] <math>\mu</math> to be a probability measure on a [[σ-algebra]] are that:

* <math>\mu</math> must return results in the [[unit interval]] <math>[0, 1],</math> returning <math>0</math> for the empty set and <math>1</math> for the entire space.
* <math>\mu</math> must satisfy the ''[[Sigma-additive set function|countable additivity]]'' property that for all [[countable]] collections <math>E_1, E_2, \ldots</math> of pairwise [[disjoint sets]]: <math display=block> \mu\left(\bigcup_{i \in \N} E_i\right) = \sum_{i \in \N} \mu(E_i).</math>

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

The [[conditional probability]] based on the intersection of events defined as:
<math display=block>\mu (B \mid A) = \frac{\mu(A \cap B)}{\mu(A)}.</math>
satisfies the probability function requirements so long as <math>\mu(A)</math> is not zero.<ref>{{Cite journal |last1=Dekking |first1=Frederik Michel |last2=Kraaikamp |first2=Cornelis |last3=Lopuhaä |first3=Hendrik Paul |last4=Meester |first4=Ludolf Erwin |date=2005 |title=A Modern Introduction to Probability and Statistics |url=https://link.springer.com/book/10.1007/1-84628-168-7 |journal=Springer Texts in Statistics |language=en |doi=10.1007/1-84628-168-7 |isbn=978-1-85233-896-1 |issn=1431-875X|url-access=subscription }}</ref><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 <math>1,</math> and the additive property is replaced by an order relation based on [[set inclusion]].

==Example applications==

In many cases, [[statistical physics]] uses ''probability measures'', but not all [[measure theory|measures]] it uses are probability measures.{{clarify|reason=this sentence make no sense on its own, see Talk page|date=May 2025}}<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> 

''Market measures'' which assign probabilities to [[financial market]] spaces based on observed market movements are examples of probability measures which are of interest in [[mathematical finance]]; for example, 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 statistical physics, for 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>

[[Ultrafilter]]s can be understood as <math>\{0, 1\}</math>-valued probability measures, allowing for many intuitive proofs based upon measures. For instance, [[Hindman’s theorem|Hindman's Theorem]] can be proven from the further investigation of these measures, and their [[convolution]] in particular.

==See also==

* {{annotated link|Borel measure}}
* {{annotated link|Fuzzy measure}}
* {{annotated link|Haar measure}}
* {{annotated link|Lebesgue measure}}
* {{annotated link|Martingale measure}}
* {{annotated link|Set function}}
* [[Probability distribution]]

==References==
{{reflist}}

==Further reading==
* {{cite book |title=Probability and Measure |first=Patrick |last=Billingsley |author-link=Patrick Billingsley |year=1995 |publisher=John Wiley |isbn=0-471-00710-2 }}
* {{cite book |title=Probability & Measure Theory |first1=Robert B. |last1=Ash |first2=Catherine A. |last2=Doléans-Dade |year=1999 |publisher=Academic Press |isbn= 0-12-065202-1}}
* [https://math.stackexchange.com/q/1073744/29780 Distinguishing probability measure, function and distribution], Math Stack Exchange

==External links==
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[[Category:Experiment (probability theory)]]
[[Category:Measures (measure theory)]]

[[pl:Miara probabilistyczna]]