Cylindrical measure
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
.
References
[1] | I.M. Gel'fand, N.Ya. Vilenkin, "Generalized functions. Applications of harmonic analysis" , 4 , Acad. Press (1968) (Translated from Russian) |
[2] | A.G. Vitushkin, "On higher-dimensional variations" , Moscow (1955) (In Russian) |
Comments
Concerning the -dimensional variation of a function see Variation of a function.
References
[a1] | L. Schwartz, "Radon measures on arbitrary topological spaces and cylindrical measures" , Oxford Univ. Press (1973) |
Cylindrical measure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cylindrical_measure&oldid=11563