Vitali variation

One of the numerical characteristics of a function of several variables. It may be considered as a multi-dimensional analogue of the variation of a function of one variable. Let the function be defined on an -dimensional parallelepipedon . One introduces the following notation:

Let be an arbitrary subdivision of the parallelepipedon by hyperplanes

into -dimensional parallelepipeda. Let be the least upper bound of sums of the type

(*)

taken over all possible subdivisions of . If , one says that the function has bounded (finite) Vitali variation on , while the class of all such functions is denoted by or simply by . The class was defined by G. Vitali [1]. The same definition of variation was subsequently proposed by H. Lebesgue [2] and M. Fréchet [3]. A real-valued function , defined on , belongs to the class if and only if it can be represented in the form , where the functions and are such that, for each of them, the sums of the type (*), taken without the modulus sign, are non-negative [4] (the analogue of the Jordan decomposition of a function of bounded variation of one variable). The functions of class may be used to introduce the multi-dimensional Stieltjes integral. In particular, for any function which is continuous on and any function of class the integral exists [3].

References

[1]	G. Vitali, "Sui gruppi di punti e sulle funzioni di variabili reali" Atti Accad. Sci. Torino , 43 (1908) pp. 75–92
[2]	H. Lebesgue, "Sur l'intégration des fonctions discontinues" Ann. Sci. École Norm. Sup. (3) , 27 (1910) pp. 361–450
[3]	M. Fréchet, "Extension au cas d'intégrales multiples d'une définition de l'intégrale due à Stieltjes" Nouv. Ann. Math. ser. 4 , 10 (1910) pp. 241–256
[4]	H. Hahn, "Theorie der reellen Funktionen" , 1 , Springer (1921)
[5]	F. Riesz, B. Szökefalvi-Nagy, "Functional analysis" , F. Ungar (1955) (Translated from French)