View source for Measure in a topological vector space

<!--
m0632501.png
$#A+1 = 41 n = 0
$#C+1 = 41 : ~/encyclopedia/old_files/data/M063/M.0603250 Measure in a topological vector space
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

A term used to designate a measure given in a [[Topological vector space|topological vector space]] when one wishes to stress those properties of the measure that are connected with the linear and topological structure of this space. A general problem encountered in the construction of a measure in a topological vector space is that of extending a [[Pre-measure|pre-measure]] to a measure. Let  $  E $
be a (real or complex) locally convex space and let  $  \mathfrak A ( E) $
be the algebra of its cylindrical sets (cf. [[Cylinder set|Cylinder set]]). Suppose that a pre-measure is defined on  $  \mathfrak A ( E) $. 
It is required to extend this pre-measure to a countably-additive measure defined on the  $  \sigma $-
algebra  $  \mathfrak B _ {0} ( E) $— 
the smallest  $  \sigma $-
algebra containing  $  \mathfrak A ( E) $. 
$  \mathfrak B _ {0} ( E) $
is the smallest of all  $  \sigma $-
algebras (weakly Borel, Borel, etc.) that are naturally connected with the topology of  $  E $; 
for a large class of spaces  $  E $
these  $  \sigma $-
algebras coincide. In the particular, yet most important, case when the space  $  E = V  ^  \prime  $, 
i.e. it is the dual of some locally convex space  $  V $, 
endowed with the weak-* topology (so that  $  E  ^  \prime  = V $), 
in order that a pre-measure  $  \mu $
in  $  V  ^  \prime  $
admits an extension to a measure, it suffices that its characteristic functional (Fourier transform)

$$ 
\chi ( \phi )  =  \int\limits _ {V  ^  \prime  }  \mathop{\rm exp} \{ ix( \phi ) \}  d
\mu ( x),\ \ 
x \in V  ^  \prime  ,\ \ 
\phi \in V ,
$$

be continuous in the so-called Sazonov topology on the space  $  V $(
i.e. in the topology generated by all continuous Hilbert semi-norms in  $  V $) 
and in a number of cases — for example, if  $  V $
is a [[Fréchet space|Fréchet space]] — it is necessary that the characteristic functional be continuous in the original topology of  $  V $. 
For instance, if  $  V $
is a [[Nuclear space|nuclear space]], the Sazonov topology is identical with the original topology, and every pre-measure in  $  V  ^  \prime  $
with a continuous characteristic functional extends to a measure. In the case of a pre-measure defined on a Hilbert space  $  H $
the sufficient condition for its extendability to a measure formulated above is also necessary. In addition to this general criterion for the extendability of pre-measures to measures, partial results of this type, applicable to specific classes of measures (or classes of spaces), are available. For example, a Gaussian pre-measure on  $  V  ^  \prime  $, 
where  $  V  ^  \prime  $
is a locally convex space (i.e. a pre-measure the restriction of which to any  $  \sigma $-
algebra  $  \mathfrak B ( L) \subset  \mathfrak A ( V  ^  \prime  ) $, 
$  L \subset  V  ^  \prime  $, 
$   \mathop{\rm dim}  L < \infty $, 
is a Gaussian distribution with correlation functional  $  B( \phi _ {1} , \phi _ {2} ) $, 
$  \phi _ {1} , \phi _ {2} \in V $) 
extends to a measure if there exists a convex neighbourhood of zero in  $  V $
with [[Epsilon-entropy| $  \epsilon $-
entropy]], in the metric defined by the inner product  $  \langle  \phi _ {1} , \phi _ {2} \rangle = B( \phi _ {1} , \phi _ {2} ) $, 
smaller than two.

For a sequence of (probability) measures in a dual space  $  V  ^  \prime  $
to converge weakly it suffices that the characteristic functionals of these measures converge pointwise (this condition is also necessary) and that they be equicontinuous at zero in the Sazonov topology in  $  V $, 
and it is necessary that these functionals be equicontinuous in the original topology of  $  V $. 
In the case of a Hilbert space  $  V $, 
conditions necessary and sufficient for the weak compactness of a family of measures in  $  V $
are known. These are also expressible in terms of their characteristic functionals. The following aspects have been investigated (1982) only for Gaussian measures: the problem of the quasi-invariance of a measure in a topological vector space (see [[Quasi-invariant measure|Quasi-invariant measure]]) with respect to some set of translations (the set of quasi-invariance) of this space (it is known that for a number of infinite-dimensional vector spaces the set of quasi-invariance of a non-zero measure does not necessarily coincide with the whole space); and criteria for the absolute continuity of one measure with respect to another. The study of measures in topological vector spaces is mainly connected with integrals over trajectories (cf. [[Integral over trajectories|Integral over trajectories]]), and also with the theory of generalized random fields, and is to a high degree stimulated by the applications of these theories in physics and mechanics.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Dunford,   J.T. Schwartz,   "Linear operators. General theory" , '''1''' , Interscience  (1958)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N. Bourbaki,   "Elements of mathematics. Integration" , Addison-Wesley  (1975)  pp. Chapt.6;7;8  (Translated from French)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I.I. [I.I. Gikhman] Gihman,   A.V. [A.V. Skorokhod] Skorohod,   "The theory of stochastic processes" , '''1''' , Springer  (1974)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  I.M. Gel'fand,   N.Ya. Vilenkin,   "Generalized functions. Applications of harmonic analysis" , '''4''' , Acad. Press  (1968)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  V.N. Sudakov,   "Geometric problems in the theory of infinite-dimensional probability distributions"  ''Proc. Steklov Inst. Math.'' , '''141'''  (1976)  ''Trudy Mat. Inst. Steklov.'' , '''141'''  (1976)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  O.G. Smolyanov,   S.V. Fomin,   "Measures on linear topological spaces"  ''Russian Math. Surveys'' , '''31''' :  4  (1976)  pp. 1–53  ''Uspekhi Mat. Nauk'' , '''31''' :  4  (1976)  pp. 3–56</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L. Schwartz,   "Radon measures on arbitrary topological spaces and cylindrical measures" , Oxford Univ. Press  (1973)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  N.N. Vakhania,   "Probability distributions on linear spaces" , North-Holland  (1981)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  N.N. Vakhania,   V.I. Tarieladze,   S.A. Chobanyan,   "Probability distributions on Banach spaces" , Reidel  (1987)  (Translated from Russian)</TD></TR></table>