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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as ''countable additivity''. The difference between a probability measure and the more general notion of measure (which includes concepts like
area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an op ...
or
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). ...
) 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; for example, the value assigned to "1 or 2" in a throw of a dice 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

The requirements for a function \mu to be a probability measure on a probability space are that: * \mu must return results in the
unit interval In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted ' (capital letter ). In addition to its role in real analysis ...
, 1 returning 0 for the empty set and 1 for the entire space. * \mu must satisfy the ''countable additivity'' property that for all countable collections \ of pairwise disjoint sets: \mu\left(\bigcup_ E_i\right) = \sum_ \mu(E_i). For example, given three elements 1, 2 and 3 with probabilities 1/4, 1/4 and 1/2, the value assigned to \ 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: \mu (B \mid A) = \frac. satisfies the probability measure requirements so long as \mu(A) is not zero. Probability measures are distinct from the more general notion of 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 A financial market is a market in which people trade financial securities and derivatives at low transaction costs. Some of the securities include stocks and bonds, raw materials and precious metals, which are known in the financial ma ...
spaces based on actual market movements are examples of probability measures which are of interest in mathematical finance; for example, in the pricing of
financial derivative In finance, a derivative is a contract that ''derives'' its value from the performance of an underlying entity. This underlying entity can be an asset, index, or interest rate, and is often simply called the "underlying". Derivatives can be u ...
s. For instance, a risk-neutral measure is a probability measure which assumes that the current value of assets is the
expected value In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a ...
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. 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. 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 under congruence, although it may do so in the case of systems with just one degree of freedom. Probability measures are also used in mathematical biology. 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 Amino acids are organic compounds that contain both amino and carboxylic acid functional groups. Although hundreds of amino acids exist in nature, by far the most important are the alpha-amino acids, which comprise proteins. Only 22 alpha ...
in a sequence.''Discovering biomolecular mechanisms with computational biology'' by Frank Eisenhaber 2006
page 127
/ref> Ultrafilters can be understood as \-valued probability measures, allowing for many intuitive proofs based upon measures. For instance, Hindman's Theorem can be proven from the further investigation of these measures, and their
convolution In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' ...
in particular.


See also

* * * * * *


References


Further reading

* *


External links

* {{Authority control Experiment (probability theory) Measures (measure theory) pl:Miara probabilistyczna