# Probability measure

*probability distribution, probability*

A real non-negative function on a class of subsets (events) of a non-empty set (the space of elementary events) forming a -field (i.e. a set closed with respect to countable set-theoretic operations) such that

if for (-additivity).

### Examples of probability measures.

1) ; is the class of all subsets of ; (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) ; is the class of all subsets of ;

where (the Poisson distribution);

3) ; is the class of Borel subsets of ;

(the normal distribution);

4) is the space of continuous real functions on that vanish at the point zero; is the class of Borel subsets with respect to the topology of uniform convergence; is the measure which is uniquely defined by the formula

where is an arbitrary natural number and (the Wiener measure).

#### References

[1] | A.N. Kolmogorov, "Foundations of the theory of probability" , Chelsea, reprint (1950) (Translated from Russian) |

[2] | B.V. Gnedenko, "The theory of probability" , Chelsea, reprint (1962) (Translated from Russian) |

#### Comments

#### References

[a1] | P. Billingsley, "Probability and measure" , Wiley (1979) |

**How to Cite This Entry:**

Probability measure.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Probability_measure&oldid=11376