Abstract
Full Text
UDC 517.397
MATHEMATICS
G. E. SHILOV
ON MEASURES IN LINEAR SPACES
(Presented by Academician A. N. Kolmogorov on 26 X 1965)
Let \(X\) be a real linear space and let \(X'\) be some linear system of linear functionals on \(X\). A natural number \(n\), a collection of \(n\) functionals \(f_1,\ldots,f_n\) from \(X'\), and a Borel set \(A\) in the real \(n\)-dimensional space \(R^n\) determine in \(X\) the cylindrical set
\[ C = C(f_1,\ldots,f_n,A)=\{x\in X: [(f_1,x),\ldots,(f_n,x)]\in A\} \tag{1} \]
with base \(A\) and generators \(f_1,\ldots,f_n\).
We shall say that a premeasure is given on the totality \(\mathfrak A\) of all cylindrical sets if to each set \(C\in\mathfrak A\) there is assigned a nonnegative number \(\mu C\) in such a way that: a) \(\mu C\) depends only on the set \(C\) itself and not on its representation in the form (1); b) on the totality \(\mathfrak A_f\) of all cylindrical sets \(C\in\mathfrak A\) with fixed collection \(f=(f_1,\ldots,f_n)\), the function \(\mu C\) is countably additive (countable additivity on all of \(\mathfrak A\) is not assumed); c) \(\mu X=1\).
We pose the question under what conditions a premeasure \(\mu\) can be extended to a measure—a countably additive set function defined on a \(\sigma\)-ring of subsets \(E\subset X\) containing all cylindrical sets.
If \(X\) is the space of all real functions \(x=x(t)\) on some set \(T\), and the functionals \(f\in X'\) are generated by the values of \(x\) at individual points \(t\in T\), then a positive answer is the content of a theorem of A. N. Kolmogorov \(\left({}^{1}\right)\). There are a number of conditions due to various authors that clarify this question under certain assumptions concerning the existence and properties of a topology on \(X\) \(\left({}^{2,3}\right)\). The following simple result pertains to spaces without topology. Suppose that \(X'\) is a total system of functionals on \(X\), i.e., from the equalities \((f,x)=0\), satisfied for a fixed \(x\in X\) for all \(f\in X'\), it follows that \(x=0\). Then \(X\) can be regarded as a space of real functions \(x(f)=(f,x)\), where \(f\) ranges over \(X'\) or even only over some total subset \(T\) in \(X'\). Let \(\hat X\) be the space of all real functions \(x=x(f)\), \(f\in T\), with the premeasure of cylindrical sets (with functionals \(f\), equal to the values of the functions \(x\) at the points \(f\in T\)) given by the formula
\[ \hat\mu\{x\in\hat X:[x(f_1),\ldots,x(f_n)]\in A\} = \mu\{x\in X:[(f_1,x),\ldots,(f_n,x)]\in A\}. \]
By Kolmogorov’s theorem, the premeasure \(\hat\mu\) is completed in \(\hat X\) to a countably additive measure defined on some \(\sigma\)-ring of subsets of the set \(\hat X\).
Theorem 1. The premeasure \(\mu\) is completed in \(X\) to a measure if and only if the image of \(X\) under the mapping \(x\to x(f)\in\hat X\) fills, in the space \(\hat X\), a set of outer \(\hat\mu\)-measure \(1\).
A premeasure \(\mu\) is called Gaussian if to each collection of functionals \(f=(f_1,\ldots,f_n)\) there correspond a positive quadratic form \(Q_f(\xi,\xi)\), \(\xi\in R^n\), and a constant \(c_f\) such that, for any Borel set \(A\subset R^n\),
\[ \mu C(f_1,\ldots,f_n,A)=c_f \int_A \exp[-Q_f(\xi,\xi)]\,d\xi_1\cdots d\xi_n. \]
If, for the space \(X\) with Gaussian premeasure \(\mu\), a countable subsystem \(g_1, g_2,\ldots\) has been selected in \(X'\), then, applying the orthogonalization process to it, one can pass to a new countable subsystem \(e_1,e_2,\ldots\) in \(X'\) (with the same linear hull), for which
\[ \mu C(e_1,\ldots,e_n,A)=\frac{1}{\sqrt{\pi^n}}\int_A \exp\left[-\sum_1^n \xi_k^2\right]\,d\xi_1\ldots d\xi_n . \]
If the system \(g_1,g_2,\ldots\) is total, then the system \(e_1,e_2,\ldots\) is also total; in this case the space \(\hat X\) is the space \(\Omega\) of all real sequences \(x=(x_1,x_2,\ldots)\) with the canonical Gaussian measure
\[ \omega\{x\in\Omega:(x_1,\ldots,x_n)\in A\} =\frac{1}{\sqrt{\pi^n}}\int_A \exp\left[-\sum_1^n \xi_k^2\right]\,d\xi_1\ldots d\xi_n . \]
By virtue of Theorem 1, the Gaussian premeasure \(\mu\) extends on \(X\) to a measure if and only if the image of \(X\) under the mapping
\[
x\to [(e_1,x),(e_2,x),\ldots]\in\Omega
\]
fills in \(\Omega\) a set of outer measure 1. Therefore, for applications it is essential to have a sufficiently broad supply of subsets of \(\Omega\) of full measure. We give several examples (the original formulation of which was given earlier in the language of probability theory):
Example 1 (B. V. Gnedenko \((^4)\)).
\[ \omega\left\{x\in\Omega:\lim_{n\to\infty}\frac{|x_n|}{\sqrt{\ln n}}=1\right\}=1. \]
Example 2 (A. N. Kolmogorov \((^5)\)). Let a function \(g(\xi)\), \(-\infty<\xi<\infty\), be given such that
\[ \frac{1}{\sqrt{\pi}}\int_{-\infty}^{\infty} g(\xi)e^{-\xi^2}\,d\xi=a,\qquad \frac{1}{\sqrt{\pi}}\int_{-\infty}^{\infty} g^2(\xi)e^{-\xi^2}\,d\xi<\infty . \]
Then
\[ \omega\left\{x\in\Omega:\lim_{n\to\infty}\frac{1}{n}\sum_1^n g(x_k)=a\right\}=1. \]
Example 3 (A. N. Kolmogorov and A. Ya. Khinchin \((^6)\)). Let positive numbers \(a_1,a_2,\ldots\) be given; set
\[ E_{\langle a_n\rangle}=\left\{x\in\Omega:\sum_1^\infty a_n x_n^2<\infty\right\}. \]
Then
\[ \omega E_{\langle a_n\rangle}= \begin{cases} 1, & \text{if } \displaystyle\sum_1^\infty a_n<\infty,\\[6pt] 0, & \text{if } \displaystyle\sum_1^\infty a_n=\infty. \end{cases} \]
Let \(f_1,f_2,\ldots\) be a numerical sequence. By Kolmogorov’s theorem \((^7)\), the series
\[
[f,x]\equiv\sum_1^\infty f_n x_n
\]
converges on \(\Omega\) in the mean square
(and also almost everywhere) if and only if \(\sum_1^\infty f_n^2<\infty\). In this case
\[ \omega\{x\in\Omega:[f,x]>c\} = \frac{1}{\sqrt{\pi}} \int_{c/\left(\sum_1^\infty f_n^2\right)^{1/2}}^\infty e^{-\xi^2}\,d\xi . \]
The expression \([f,x]\) is the general form of a linear measurable functional on \(\Omega\) \((^8)\). If several sequences \(f_1^j, f_2^j,\ldots\) \((j=1,\ldots,m)\), orthogonal and normalized in the space \(l_2\), are given, then
\[ \omega\{x\in\Omega:([f^1,x],\ldots,[f^m,x])\in A\} = \frac{1}{\sqrt{\pi^m}}\int_A \exp\left[-\sum_1^m \xi_k^2\right]\,d\xi_1\ldots d\xi_m . \]
It follows that a linear transformation of the space \(\Omega\) into itself, given by the formulas
\[ y_n=\sum_{m=1}^{\infty} u_{mn}x_m \]
with an orthogonal matrix \(U=\|u_{mn}\|\), carries measurable sets \(E\subset\Omega\) again into measurable sets of the same measure. Fan Dyk Tinh showed in \((^8)\) that the indicated transformation is the general form of a measurable linear transformation in \(\Omega\) preserving measure. There he also indicated the general form of linear transformations \(y=Bx\) that carry measurable sets into measurable sets and do not preserve measure, but have an absolutely continuous set function \(\omega(BE)\).
Wiener measure. Let \(H\) be a separable Hilbert space and \(e_1,e_2,\ldots\) an orthonormal basis in \(H\). Each vector \(z\in H\) is represented by a sequence of numbers \(z_n=(e_n,z)\) with \(\sum_1^\infty z_n^2<\infty\). Fix a sequence of positive numbers \(\lambda_1,\lambda_2,\ldots\) and associate with the point \(z\in H\) the point \((\lambda_1z_1,\lambda_2z_2,\ldots)\) of the space \(\Omega\). Introduce in \(H\) the premeasure
\[ \mu\{z\in H:(z_1,\ldots,z_n)\in A\subset R_n\} = \omega\{x\in\Omega:(\lambda_1z_1,\ldots,\lambda_nz_n)\in A\}. \]
The premeasure \(\mu\) is extended in \(H\) to a measure if the image of \(H\) in \(\Omega\) fills a set of full measure (Theorem 1). By the Kolmogorov–Khinchin criterion \((^3)\), this condition is satisfied if and only if the series \(\sum_1^\infty \lambda_n^{-2}\) converges.
The measure in the space \(H\) obtained for \(\lambda_n\equiv n\) will be called the abstract Wiener measure. If as \(H\) one takes the space \(L_2(0,\pi)\) of all square-integrable functions \(z(t)\) on the interval \([0,\pi]\), orthogonal to 1, and as basis vectors \(e_n\) takes the functions
\[ e_n(t)=\sqrt{\frac{2}{\pi}}\cos nt, \]
then we obtain the classical Wiener measure. The functions \(z(t)\in \bar L_2(0,\pi)\) that are continuous and satisfy the Hölder condition \(|z(t')-z(t'')|\le C|t'-t''|^\alpha\) with any exponent \(\alpha<1/2\) fill in \(\bar L_2(0,\pi)\) a set \(W\) of full measure. If from the functions \(z(t)\in W\) we pass to the functions \(\varphi(t)=z(t)-z(0)\), equal to 0 for \(t=0\), we obtain the second classical realization of Wiener measure \((^9)\). Kolmogorov’s criterion 2, after a certain modification, leads to a set of full measure (indicated by Cameron and Martin \((^{10})\) for the special case \(g(\xi)=\xi^2\)):
\[ \omega\left\{\varphi(t):\lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^{n} g\left[ \frac{\varphi\left(\frac{k}{n}\pi\right)-\varphi\left(\frac{k-1}{n}\pi\right)} {\sqrt{\pi/n}} \right]=a\right\}=1. \]
Moscow State University
named after M. V. Lomonosov
Received
28 IX 1965
REFERENCES
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