# Probability space

2010 Mathematics Subject Classification: Primary: 60A05 [MSN][ZBL]


1. $\def\a{\alpha}\P_{t_1,\dots,t_n}(l_{y_1,\dots,y_n}) = \P_{t_{\a_1},\dots,t_{\a_n}}(l_{y_{\a_1},\dots,y_{\a_n}})$ for all $(y_1,\dots,y_n) \in \R^n$, where $l_{y_1,\dots,y_n} = \{x = (x_1,\dots,x_n)\;:\; x_i\le y_i,\ i=1,\dots,n\}$ and $\a_1,\dots,\a_n$ is an arbitrary rearrangement of the numbers $1,\dots,n$;
2. $\P_{t_1,\dots,t_n}(l_{y_1,\dots,y_{n-1},\infty}) = \P_{t_1,\dots,t_{n-1}}(l_{y_1,\dots,y_{n-1}})$.

Then there exists a probability measure $\P$ on the smallest $\sigma$-algebra $\A$ of subsets of the product $\R^T = \{x = \{x_t\}\;:\ t\in T,\;x_t\in \R^1\}$ with respect to which all the coordinate functions $t(x) = x_t$ are measurable, such that for any finite subset $t_1,\dots,t_n$ of $T$ and for any $n$-dimensional Borel set $B$ the following equation is true: $$\P_{t_1,\dots,t_n}(B) = \P\{x\in R^T\;:\;t_1(x),\dots,t_n(x) \in B \}.$$

#### References

 [Bi] P. Billingsley, "Probability and measure", Wiley (1979) MR0534323 Zbl 0411.60001 [GnKo] B.V. Gnedenko, A.N. Kolmogorov, "Limit distributions for sums of independent random variables", Addison-Wesley (1954) (Translated from Russian) MR0062975 Zbl 0056.36001 [Ko] A.N. Kolmogorov, "Foundations of the theory of probability", Chelsea, reprint (1950) (Translated from Russian) MR0032961 Zbl 0074.12202 [Ne] J. Neveu, "Mathematical foundations of the calculus of probabilities", Holden-Day (1965) (Translated from French) MR0198505
How to Cite This Entry:
Probability space. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Probability_space&oldid=29731
This article was adapted from an original article by V.V. Sazonov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article