Probability space

2020 Mathematics Subject Classification: Primary: 60A05 [MSN][ZBL]

$ \newcommand{\R}{\mathbb R} \newcommand{\Om}{\Omega} \newcommand{\A}{\mathcal A} \newcommand{\P}{\mathbf P} $ A probability space (or also probability field) is a triple $(\Om,\A,\P)$ consisting of a non-empty set $\Om$, a class $\A$ of subsets of $\Om$ which is a σ-algebra (i.e. is closed with respect to the set-theoretic operations executed a countable number of times) and a probability measure $\P$ on $\A$. The concept of a probability space is due to A.N. Kolmogorov [Ko]. The points of $\Om$ are said to be elementary events, while the set $\Om$ itself is referred to as the space of elementary events or the sample space. The subsets of $\Om$ belonging to $\A$ are (random) events. The study of probability spaces is often restricted to the study of complete probability spaces, i.e. spaces which satisfy the requirement $B\in\A$, $A\subset B$, $\P(B)=0$ implies $A\in\A$. If $(\Om,\A,\P)$ is an arbitrary probability space, the class of sets of the type $A\cup N$, where $A\in\A $ and $N\subset M$, for some $M\in\A$ with $\P(M)=0$, forms a σ-algebra $\overline{\A}$, while the function $\overline{\P}$ on $\overline{\A}$ defined by the formula $\overline{\P}(A\cup N)=\P(A)$ is a probability measure on $\A$. The space $(\Om,\overline{\A},\overline{\P})$ is complete and is said to be the completion of $(\Om,\A,\P)$. Usually one may restrict attention to perfect probability spaces, i.e. spaces such that for any real $\A$-measurable function $f$ and any set $E$ on the real line for which $f^{-1}(E)\in\A$, there exists a Borel set $B$ such that $B\subset E$ and $\P(f^{-1}(E))=\P(f^{-1}(B))$. Certain "pathological" effects (connected with the existence of conditional probabilities, the definition of independent random variables, etc.), which occur in the general scheme, cannot occur in perfect probability spaces. The problem of the existence of probability spaces satisfying some given special requirements is not trivial in many cases. One result of this type is the fundamental Kolmogorov consistency theorem: Let to each ordered $n$-tuple $t_1,\dots,t_n$ of elements of a set $T$ correspond a probability measure $\P_{t_1,\dots,t_n}$ on the Borel sets of the Euclidean space $\R^n$ and let the following consistency conditions be satisfied:

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