# Probability measure

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probability distribution, probability

2010 Mathematics Subject Classification: Primary: 60-01 [MSN][ZBL]

A real non-negative function ${\mathsf P}$ on a class ${\mathcal A}$ of subsets (events) of a non-empty set $\Omega$ (the space of elementary events) forming a $\sigma$- field (i.e. a set closed with respect to countable set-theoretic operations) such that

$${\mathsf P} ( \Omega ) = 1 \ \textrm{ and } \ \ {\mathsf P} \left ( \cup _ { i=1} ^ \infty A _ {i} \right ) = \ \sum _ { i=1 } ^ \infty {\mathsf P} ( A _ {i} )$$

if $A _ {i} \cap A _ {j} = \emptyset$ for $i \neq j$( $\sigma$- additivity).

## Contents

### Examples of probability measures.

1) $\Omega = \{ 1, 2 \}$; ${\mathcal A}$ is the class of all subsets of $\Omega$; ${\mathsf P} ( \{ 1 \} ) = {\mathsf P} ( \{ 2 \} ) = 1 / 2$ (this probability measure corresponds to a random experiment consisting in throwing a symmetrical coin; if heads correspond to 1 while tails correspond to 2, the probability of throwing heads (tails) is 1/2);

2) $\Omega = \{ 0, 1 , . . . \}$; ${\mathcal A}$ is the class of all subsets of $\Omega$;

$${\mathsf P} ( \{ k \} ) = \ \frac{\lambda ^ {k} }{k!} e ^ {- \lambda } ,$$

where $\lambda > 0$ (the Poisson distribution);

3) $\Omega = \mathbf R ^ {1}$; ${\mathcal A}$ is the class of Borel subsets of $\mathbf R ^ {1}$;

$${\mathsf P} ( A) = \ \frac{1}{\sqrt {2 \pi } } \int\limits _ { A } e ^ {- x ^ {2} /2 } dx$$ (the normal distribution);

4) $\Omega = C _ {0} [ 0, 1]$ is the space of continuous real functions $x( t)$ on $[ 0, 1]$ that vanish at the point zero; ${\mathcal A}$ is the class of Borel subsets with respect to the topology of uniform convergence; ${\mathsf P}$ is the measure which is uniquely defined by the formula

$${\mathsf P} ( x : a _ {i} < x ( t _ {i} ) < b _ {i} ,\ i = 1 \dots n) =$$

$$= \ ( 2 \pi ) ^ {- n/2} \prod _ { i=1 } ^ { n } ( t _ {i} - t _ {i-1} ) ^ {- 1/2} \times$$

$$\times \int\limits _ { a _ {1} } ^ { {b _ 1 } } \dots \int\limits _ {a _ {n} } ^ { {b _ n} } \mathop{\rm exp} \left \{ - \frac{1}{2} \sum _ { i=1 } ^ { n } \frac{( x _ {i} - x _ {i-1} ) ^ {2} }{t _ {i} - t _ {i-1} } \right \} dx _ {1} \dots d x _ {n} ,$$

where $n$ is an arbitrary natural number and $0 = t _ {0} < t _ {1} < \dots < t _ {n} \leq 1$ (the Wiener measure).

#### References

 [K] A.N. Kolmogorov, "Foundations of the theory of probability" , Chelsea, reprint (1950) (Translated from Russian) MR0032961 [G] B.V. Gnedenko, "The theory of probability", Chelsea, reprint (1962) (Translated from Russian)

#### References

 [B] P. Billingsley, "Probability and measure" , Wiley (1979) MR0534323 Zbl 0411.60001
How to Cite This Entry:
Probability measure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Probability_measure&oldid=51252
This article was adapted from an original article by V.V. Sazonov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article