# 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 and 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. , ). 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 . If the function is continuous on the rectangle , then the surface has finite area if and only if belongs to (cf. Tonelli theorem).

How to Cite This Entry:
Tonelli plane variation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tonelli_plane_variation&oldid=16914
This article was adapted from an original article by B.I. Golubov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article