Probability measure
Encyclopedia
In mathematics, a probability measure is a real-valued function
Real-valued function
In mathematics, a real-valued function is a function that associates to every element of the domain a real number in the image....

defined on a set of events in a probability space
Probability space
In probability theory, a probability space or a probability triple is a mathematical construct that models a real-world process consisting of states that occur randomly. A probability space is constructed with a specific kind of situation or experiment in mind...

that satisfies measure
Measure (mathematics)
In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...

properties such as countable additivity. The difference between a probability measure and the more general notion of measure (which includes concepts like area
Area
Area is a quantity that expresses the extent of a two-dimensional surface or shape in the plane. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat...

or volume
Volume
Volume is the quantity of three-dimensional space enclosed by some closed boundary, for example, the space that a substance or shape occupies or contains....

) is that a probability measure must assign 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

The requirements for a function to be a probability measure on a probability space
Probability space
In probability theory, a probability space or a probability triple is a mathematical construct that models a real-world process consisting of states that occur randomly. A probability space is constructed with a specific kind of situation or experiment in mind...

are that:
must return results in the unit interval
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...

[0, 1], returning 0 for the empty set and 1 for the entire space.
must satisfy the countable additivity property that for all countable collections of pairwise disjoint sets:

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
Conditional probability
In probability theory, the "conditional probability of A given B" is the probability of A if B is known to occur. It is commonly notated P, and sometimes P_B. P can be visualised as the probability of event A when the sample space is restricted to event B...

based on the intersection of events defined as:

satisfies the probability measure requirements so long as is not zero.

Probability measures are distinct from the more general notion of fuzzy measures
Fuzzy measure theory
Fuzzy measure theory considers a number of special classes of measures, each of which is characterized by a special property. Some of the measures used in this theory are plausibility and belief measures, fuzzy set membership function and the classical probability 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
Financial market
In economics, a financial market is a mechanism that allows people and entities to buy and sell financial securities , commodities , and other fungible items of value at low transaction costs and at prices that reflect supply and demand.Both general markets and...

spaces based on actual market movements are examples of probability measures which are of interest in mathematical finance
Mathematical finance
Mathematical finance is a field of applied mathematics, concerned with financial markets. The subject has a close relationship with the discipline of financial economics, which is concerned with much of the underlying theory. Generally, mathematical finance will derive and extend the mathematical...

, e.g. in the pricing of financial derivatives. For instance, a risk-neutral measure
Risk-neutral measure
In mathematical finance, a risk-neutral measure, is a prototypical case of an equivalent martingale measure. It is heavily used in the pricing of financial derivatives due to the fundamental theorem of asset pricing, which implies that in a complete market a derivative's price is the discounted...

is a probability measure which assumes that the current value of assets is the expected value
Expected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...

of the future payoff 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
Complete market
In economics, a complete market is one in which the complete set of possible gambles on future states-of-the-world can be constructed with existing assets without friction. Every agent is able to exchange every good, directly or indirectly, with every other agent without transaction costs...

.

Not all measures that intuitively represent chance or likelihood are probability measures. For instance, although the fundamental concept of a system in statistical mechanics
Statistical mechanics
Statistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...

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
Congruence relation
In abstract algebra, a congruence relation is an equivalence relation on an algebraic structure that is compatible with the structure...

, although it may do so in the case of systems with just one degree of freedom.

Probability measures are also used in mathematical biology
Mathematical biology
Mathematical and theoretical biology is an interdisciplinary scientific research field with a range of applications in biology, medicine and biotechnology...

. For instance, in comparative sequence analysis
Sequence analysis
In bioinformatics, the term sequence analysis refers to the process of subjecting a DNA, RNA or peptide sequence to any of a wide range of analytical methods to understand its features, function, structure, or evolution. Methodologies used include sequence alignment, searches against biological...

a probability measure may be defined for the likelihood that a variant may be permissible for an amino acid
Amino acid
Amino acids are molecules containing an amine group, a carboxylic acid group and a side-chain that varies between different amino acids. The key elements of an amino acid are carbon, hydrogen, oxygen, and nitrogen...

in a sequence.