Difference between revisions of "Probability space"
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\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|probability measure]] $\P$ on $\A$. The concept of a probability space is due to A.N. Kolmogorov [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074960/p07496016.png" /> is an arbitrary probability space, the class of sets of the type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074960/p07496017.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074960/p07496018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074960/p07496019.png" />, for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074960/p07496020.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074960/p07496021.png" />, forms a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074960/p07496022.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074960/p07496023.png" />, while the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074960/p07496024.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074960/p07496025.png" /> defined by the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074960/p07496026.png" /> is a probability measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074960/p07496027.png" />. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074960/p07496028.png" /> is complete and is said to be the completion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074960/p07496029.png" />. Usually one may restrict attention perfect probability spaces, i.e. spaces such that for any real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074960/p07496030.png" />-measurable function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074960/p07496031.png" /> and any set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074960/p07496032.png" /> on the real line for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074960/p07496033.png" />, there exists a Borel set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074960/p07496034.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074960/p07496035.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074960/p07496036.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074960/p07496037.png" />-tuple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074960/p07496038.png" /> of elements of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074960/p07496039.png" /> correspond a probability measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074960/p07496040.png" /> on the Borel sets of the Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074960/p07496041.png" /> and let the following consistency conditions be satisfied:
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|probability measure]] $\P$ on $\A$. The concept of a probability space is due to A.N. Kolmogorov [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074960/p07496024.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074960/p07496025.png" /> defined by the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074960/p07496026.png" /> is a probability measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074960/p07496027.png" />. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074960/p07496028.png" /> is complete and is said to be the completion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074960/p07496029.png" />. Usually one may restrict attention perfect probability spaces, i.e. spaces such that for any real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074960/p07496030.png" />-measurable function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074960/p07496031.png" /> and any set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074960/p07496032.png" /> on the real line for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074960/p07496033.png" />, there exists a Borel set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074960/p07496034.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074960/p07496035.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074960/p07496036.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074960/p07496037.png" />-tuple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074960/p07496038.png" /> of elements of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074960/p07496039.png" /> correspond a probability measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074960/p07496040.png" /> on the Borel sets of the Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074960/p07496041.png" /> and let the following consistency conditions be satisfied:
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074960/p07496042.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074960/p07496043.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074960/p07496044.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074960/p07496045.png" /> is an arbitrary rearrangement of the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074960/p07496046.png" />;
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> P. Billingsley, "Probability and measure" , Wiley (1979)</TD></TR></table>
[[Category:Probability]]
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 on defined by the formula is a probability measure on . The space is complete and is said to be the completion of . 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 ;
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:
[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)
[a1] P. Billingsley, "Probability and measure" , Wiley (1979)