Tonelli plane variation

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A numerical characteristic of a function of two variables, by means of which one defines the class of functions of bounded variation in the sense of Tonelli. Suppose that is a function given on the rectangle . Assume that the functions


are Lebesgue measurable (the first on the interval , the second on ). If

then one says that the function has bounded (finite, or: is of bounded (finite)) Tonelli plane variation on the rectangle , and the class of all such functions is denoted by . This definition was proposed by L. Tonelli (cf. [1], [2]). For continuous functions, however, another characterization (in terms of the Banach indicatrix) of the class can be found in an earlier paper of S. Banach [4]. If the function is continuous on the rectangle , then the surface has finite area if and only if belongs to (cf. Tonelli theorem).


[1] L. Tonelli, "Sur la quadrature des surfaces" C.R. Acad. Sci. Paris , 182 (1926) pp. 1198–1200
[2] L. Tonelli, "Sulla quadratura delle superficie" Atti Accad. Naz. Lincei , 3 (1926) pp. 357–363; 445–450; 633–658
[3] A.G. Vitushkin, "On higher-dimensional variations" , Moscow (1955) (In Russian)
[4] S. Banach, "Sur les lignes rectifiables et les surfaces dont l'aire est finie" Fund. Math. , 7 (1925) pp. 225–236
[5] S. Saks, "Theory of the integral" , Hafner (1952) pp. 169 (Translated from French)
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Tonelli plane variation. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by B.I. Golubov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article