Cylindrical measure

A cylindrical measure in measure theory on topological vector spaces is a finitely-additive measure $ \mu $ defined on the algebra $ \mathfrak A ( E) $ of cylinder sets in a topological vector space $ E $, that is, sets of the form

$$ \tag{* } A = F _ {\phi _ {1} \dots \phi _ {n} } ^ { - 1 } ( B), $$

where $ B \in \mathfrak B ( \mathbf R ^ {n} ) $— the Borel $ \sigma $- algebra of subsets of the space $ \mathbf R ^ {n} $, $ n = 1, 2 ,\dots $; $ \phi _ {1} \dots \phi _ {n} $ are linear functionals on $ E $, and $ F _ {\phi _ {1} \dots \phi _ {n} } $ is the mapping

$$ E \rightarrow \mathbf R ^ {n} : \ x \rightarrow \{ \phi _ {1} ( x) \dots \phi _ {n} ( x) \} \in \mathbf R ^ {n} ,\ \ x \in E. $$

Here it is assumed that the restriction of $ \mu $ to any $ \sigma $- subalgebra $ \mathfrak B _ {\phi _ {1} \dots \phi _ {n} } ( E) \subset \mathfrak A ( E) $ of sets of the form (*) with a fixed collection $ ( \phi _ {1} \dots \phi _ {n} ) $ of functionals is a $ \sigma $- additive measure on $ \mathfrak B _ {\phi _ {1} \dots \phi _ {n} } $( 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 $ \sigma $- algebra $ \mathfrak B ( \mathbf R ^ {n + 1 } ) $ of the space $ \mathbf R ^ {n + 1 } $ by means of the formula

$$ \lambda ( B) = \ \lim\limits _ {\epsilon \rightarrow 0 } \ \inf _ {\begin{array}{c} \{ A \} , \\ \mathop{\rm diam} A < \epsilon \end{array} } \ \left \{ \sum l ( A) \right \} , $$

where the lower bound is taken over all finite or countable coverings of a set $ B \in \mathfrak B ( \mathbf R ^ {n+} 1 ) $ by cylinders $ A $ with spherical bases and axes parallel to the $ ( n + 1) $- st coordinate axis in $ \mathbf R ^ {n + 1 } $; here $ l ( A) $ is the $ n $- dimensional volume of an axial section of the cylinder $ A $. When $ B $ is the graph of a continuous function $ f $ of $ n $ variables defined in a domain $ G \subset \mathbf R ^ {n} $:

$$ B = \ \{ {( x _ {1} \dots x _ {n+} 1 ) } : {x _ {n+} 1 = f ( x _ {1} \dots x _ {n} ) } \} , $$

then $ \lambda ( B) $ is the same as the so-called $ n $- dimensional variation of $ f $.

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 $ n $- 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)