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Inmathematics, aprobability measure is areal-valued function defined on a set of events in aσ-algebra that satisfiesmeasure properties such ascountable additivity.^[1] The difference between a probability measure and the more general notion of measure (which includes concepts likearea orvolume) 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 die 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.

The requirements for aset function${\displaystyle \mu }$ to be a probability measure on aσ-algebra are that:

• ${\displaystyle \mu }$ must take values in theunit interval${\displaystyle [0,1],}$ including${\displaystyle 0}$ on the empty set and${\displaystyle 1}$ on the entire space.
• ${\displaystyle \mu }$ must satisfy thecountable additivity property that for allcountable collections${\displaystyle E_1,E_2,\dots }$ of pairwisedisjoint sets:$${\displaystyle \mu \left(\bigcup _{i\in \mathbb{N}}{E}_i\right)=\sum _{i\in \mathbb{N}}\mu (E_i).}$$

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

Theconditional probability based on the intersection of events defined as:$${\displaystyle \mu (B\mid A)=\frac{\mu (A\cap B)}{\mu (A)}.}$$satisfies the probability function requirements so long as${\displaystyle \mu (A)}$ is not zero.^[2][3]

Probability measures are distinct from the more general notion offuzzy measures in which there is no requirement that the fuzzy values sum up to${\displaystyle 1,}$ and the additive property is replaced by an order relation based onset inclusion.

In many cases,statistical physics usesprobability measures, but not allmeasures it uses are probability measures.^{[clarification needed][4][5]}

Market measures which assign probabilities tofinancial market spaces based on observed market movements are examples of probability measures which are of interest inmathematical finance; for example, in the pricing offinancial derivatives.^[6] For instance, arisk-neutral measure is a probability measure which assumes that the current value of assets is theexpected value of the future payoff taken with respect to that same risk neutral measure (i.e. calculated using the corresponding risk neutral density function), anddiscounted at therisk-free rate. If there is a unique probability measure that must be used to price assets in a market, then the market is called acomplete market.^[7]

Not all measures that intuitively represent chance or likelihood are probability measures. For instance, although the fundamental concept of a system instatistical mechanics is a measure space, such measures are not always probability measures.^[4] 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 measureunder congruence, although it may do so in the case of systems with just one degree of freedom.^[5]

Probability measures are also used inmathematical biology.^[8] For instance, in comparativesequence analysis a probability measure may be defined for the likelihood that a variant may be permissible for anamino acid in a sequence.^[9]

• Borel measure – Measure defined on all open sets of a topological space
• Fuzzy measure
• Haar measure – Left-invariant (or right-invariant) measure on locally compact topological group
• Lebesgue measure – Concept of area in any dimension
• Martingale measure – Probability measurePages displaying short descriptions of redirect targets
• Set function – Function from sets to numbers
• Probability distribution

• Billingsley, Patrick (1995).Probability and Measure. John Wiley.ISBN 0-471-00710-2.
• Ash, Robert B.; Doléans-Dade, Catherine A. (1999).Probability & Measure Theory. Academic Press.ISBN 0-12-065202-1.
• Distinguishing probability measure, function and distribution, Math Stack Exchange

• Media related toProbability measure at Wikimedia Commons

• Experiment (probability theory)
• Measures (measure theory)

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