# 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)

Concerning the $n$- dimensional variation of a function see Variation of a function.