Editing Probability measure (section)

==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.