A cylindrical measure in measure theory on topological vector spaces is a finitely-additive measure defined on the algebra of cylinder sets in a topological vector space , that is, sets of the form
where — the Borel -algebra of subsets of the space , ; are linear functionals on , and is the mapping
Here it is assumed that the restriction of to any -subalgebra of sets of the form (*) with a fixed collection of functionals is a -additive measure on (other names are pre-measure, quasi-measure).
In the theory of functions of several real variables a cylindrical measure is a special case of the Hausdorff measure, defined on the Borel -algebra of the space by means of the formula
where the lower bound is taken over all finite or countable coverings of a set by cylinders with spherical bases and axes parallel to the -st coordinate axis in ; here is the -dimensional volume of an axial section of the cylinder . When is the graph of a continuous function of variables defined in a domain :
then is the same as the so-called -dimensional variation of .
|||I.M. Gel'fand, N.Ya. Vilenkin, "Generalized functions. Applications of harmonic analysis" , 4 , Acad. Press (1968) (Translated from Russian)|
|||A.G. Vitushkin, "On higher-dimensional variations" , Moscow (1955) (In Russian)|
Concerning the -dimensional variation of a function see Variation of a function.
|[a1]||L. Schwartz, "Radon measures on arbitrary topological spaces and cylindrical measures" , Oxford Univ. Press (1973)|
Cylindrical measure. R.A. Minlos (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Cylindrical_measure&oldid=11563