Probability space
probability field
$
\newcommand{\Om}{\Omega}
\newcommand{\A}{\mathcal A}
\newcommand{\P}{\mathbb P}
$
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 [1]. 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 perfect probability spaces, i.e. spaces such that for any real -measurable function
and any set
on the real line for which
, there exists a Borel set
such that
and
. 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
-tuple
of elements of a set
correspond a probability measure
on the Borel sets of the Euclidean space
and let the following consistency conditions be satisfied:
1) for all
, where
and
is an arbitrary rearrangement of the numbers
;
2) .
Then there exists a probability measure on the smallest
-algebra
of subsets of the product
with respect to which all the coordinate functions
are measurable, such that for any finite subset
of
and for any
-dimensional Borel set
the following equation is true:
![]() |
References
[1] | A.N. Kolmogorov, "Foundations of the theory of probability" , Chelsea, reprint (1950) (Translated from Russian) |
[2] | B.V. Gnedenko, A.N. Kolmogorov, "Limit distributions for sums of independent random variables" , Addison-Wesley (1954) (Translated from Russian) |
[3] | J. Neveu, "Mathematical foundations of the calculus of probabilities" , Holden-Day (1965) (Translated from French) |
Comments
References
[a1] | P. Billingsley, "Probability and measure" , Wiley (1979) |
Probability space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Probability_space&oldid=19761