Namespaces
Variants
Actions

Difference between revisions of "Probability space"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m
 
(8 intermediate revisions by 2 users not shown)
Line 1: Line 1:
''probability field''
 
  
A triple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074960/p0749601.png" /> consisting of a non-empty set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074960/p0749602.png" />, a class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074960/p0749603.png" /> of subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074960/p0749604.png" /> which is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074960/p0749605.png" />-algebra (i.e. is closed with respect to the set-theoretic operations executed a countable number of times) and a [[Probability measure|probability measure]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074960/p0749606.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074960/p0749607.png" />. The concept of a probability space is due to A.N. Kolmogorov [[#References|[1]]]. The points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074960/p0749608.png" /> are said to be elementary events, while the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074960/p0749609.png" /> itself is referred to as the space of elementary events or the sample space. The subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074960/p07496010.png" /> belonging to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074960/p07496011.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074960/p07496012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074960/p07496013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074960/p07496014.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074960/p07496015.png" />. 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:
 
  
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" />;
+
{{MSC|60A05}}
 +
{{TEX|done}}
  
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074960/p07496047.png" />.
+
$ \newcommand{\R}{\mathbb R}
 
+
\newcommand{\Om}{\Omega}
Then there exists a probability measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074960/p07496048.png" /> on the smallest <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074960/p07496049.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074960/p07496050.png" /> of subsets of the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074960/p07496051.png" /> with respect to which all the coordinate functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074960/p07496052.png" /> are measurable, such that for any finite subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074960/p07496053.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074960/p07496054.png" /> and for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074960/p07496055.png" />-dimensional Borel set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074960/p07496056.png" /> the following equation is true:
+
\newcommand{\A}{\mathcal A}
 
+
\newcommand{\P}{\mathbf P} $
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074960/p07496057.png" /></td> </tr></table>
+
A ''probability space'' (or also ''probability field'') is a triple
 
+
$(\Om,\A,\P)$ consisting of a non-empty set $\Om$, a class $\A$ of
====References====
+
subsets of $\Om$ which is a σ-algebra (i.e. is closed with respect to
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.N. Kolmogorov,   "Foundations of the theory of probability" , Chelsea, reprint  (1950)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B.V. Gnedenko,   A.N. Kolmogorov,  "Limit distributions for sums of independent random variables" , Addison-Wesley  (1954) (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J. Neveu,   "Mathematical foundations of the calculus of probabilities" , Holden-Day  (1965)  (Translated from French)</TD></TR></table>
+
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
====Comments====
+
{{Cite|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:
  
 +
# $\def\a{\alpha}\P_{t_1,\dots,t_n}(l_{y_1,\dots,y_n}) =
 +
\P_{t_{\a_1},\dots,t_{\a_n}}(l_{y_{\a_1},\dots,y_{\a_n}})$ 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 $1,\dots,n$;
 +
# $\P_{t_1,\dots,t_n}(l_{y_1,\dots,y_{n-1},\infty}) =
 +
\P_{t_1,\dots,t_{n-1}}(l_{y_1,\dots,y_{n-1}})$.
  
 +
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====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"P. Billingsley,   "Probability and measure" , Wiley  (1979)</TD></TR></table>
+
{|
 +
|-
 +
|valign="top"|{{Ref|Bi}}||valign="top"| P. Billingsley, "Probability and measure", Wiley (1979)  {{MR|0534323}}  {{ZBL|0411.60001}}
 +
|-
 +
|valign="top"|{{Ref|GnKo}}||valign="top"| B.V. Gnedenko, A.N. Kolmogorov, "Limit distributions for sums of independent random variables", Addison-Wesley (1954) (Translated from Russian)  {{MR|0062975}}  {{ZBL|0056.36001}}
 +
|-
 +
|valign="top"|{{Ref|Ko}}||valign="top"| A.N. Kolmogorov, "Foundations of the theory of probability", Chelsea, reprint (1950) (Translated from Russian)  {{MR|0032961}} {{ZBL|0074.12202}}
 +
|-
 +
|valign="top"|{{Ref|Ne}}||valign="top"| J. Neveu, "Mathematical foundations of the calculus of probabilities", Holden-Day (1965) (Translated from French) {{MR|0198505}} 
 +
|-
 +
|}

Latest revision as of 17:20, 11 May 2013


2010 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
How to Cite This Entry:
Probability space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Probability_space&oldid=15765
This article was adapted from an original article by V.V. Sazonov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article