On the Total Variation of a Function of two Variables*

Abstract

The corrections detailed below are necessitated by the discovery that a certain formula, given by Saks† for the area of a continuous surface z = f(z,y), is not correct for all such surfaces without restriction. (i) In Theorem 1, on p. 294, at the end of the last sentence but one, there should be inserted the restrictive clause ‘provided that, in the case of the formula’ (2.92), the corresponding formula for the area of the surface ∑(R) is valid’. An alternative, and equivalent, restriction ensuring the validity of (2.92) would be ‘provided that f(x,y) is absolutely continuous in L. C. Young's sense‡’. In the proof of Theorem 1, at the end of § 12, on p. 310, the footnote reference to Saks should be supplemented by one to L. C. Young's paper‡. It may be noted, further, that the formula (2.91) in Theorem 1 and the formula (12.31), p. 309, for VR[f] are both valid without the new restriction. (ii) In the proof of Theorem 5, the part of a sentence appearing on p. 311, after the comma, should be replaced by the following: … and the arguments employed by Radó and Tonelli* to deal with these expressions can be adapted without trouble to show that the total variation of the rectangle-function | ηx(I)| over R is equal to the Tonelli integral W[0;f; R], and similarly, for the functionfθ; cf. § 1, that W[θ; f; R] = W[0; fθ; Rθ] equals the total variation over Rθ of |ηx(fθ; I)|. But since ηx(fθ; I) is a continuous and additive function of rectangles, and indeed (after an obvious extension of its definition) of general polygonal regions also, this total variation will be independent of the orientation of the rectangles used in the underlying dissections of Rθ: it must thus be equal to the total variation over Rθ of |ηx(fθ; Iθ)|. Now ηx(fθ; Iθ) equals the component of the vector η(I) in the direction inclined at angle − θ to the x-axis: let us denote this by ηζ(f; I). As Iθ varies in Rθ the corresponding rectangle I will vary in R, and so, altogether, W[θ;f; R] will equal the total variation over R of R of |ηζ(f; I)|. On integrating with respect to θ over [0, π], and observing that the integral of the total variation is equal to the total variation of the integral, we see that, in virtue of (1.5) and (1.6), 2VR[f] must be equal to the total variation over R of ∫ 0 π | η ζ ( f ; I ) | d θ = 2 | η ( I ) | , as required for the theorem, the last equality resulting from the definition of ηζ(f; I).