The significant cylindrical condition and a Hausdorff rectifiable set

Breckenridge, J. C.

doi:10.1090/S0002-9939-1973-0311876-4

The significant cylindrical condition and a Hausdorff rectifiable set
HTML articles powered by AMS MathViewer

by J. C. Breckenridge PDF

Proc. Amer. Math. Soc. 37 (1973), 549-552 Request permission

Abstract:

Area measures of the Geöcze type can be constructed for continuous parametric surfaces of finite Lebesgue area provided the surface satisfies the significant cylindrical condition. It is shown that requiring this condition is equivalent to requiring the Hausdorff rectifiability of a certain subset of the range of the surface.

References

J. C. Breckenridge, Cesari-Weierstrass surface integrals and lower $k$-area, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 25 (1971), 423–446. MR 315092

Geöcze k-area and significant sets

Lamberto Cesari, Surface area, Annals of Mathematics Studies, No. 35, Princeton University Press, Princeton, N. J., 1956. MR 0074500
Herbert Federer, Essential multiplicity and Lebesgue area, Proc. Nat. Acad. Sci. U.S.A. 34 (1948), 611–616. MR 27837, DOI 10.1073/pnas.34.12.611
Hebert Federer, Currents and area, Trans. Amer. Math. Soc. 98 (1961), 204–233. MR 123682, DOI 10.1090/S0002-9947-1961-0123682-0
Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York, Inc., New York, 1969. MR 0257325
Togo Nishiura, Integrals over a product variety and Fubini theorems, Rend. Circ. Mat. Palermo (2) 14 (1965), 207–236. MR 197685, DOI 10.1007/BF02847721
Togo Nishiura, Area measure and Radó’s lower area, Trans. Amer. Math. Soc. 159 (1971), 355–367. MR 281880, DOI 10.1090/S0002-9947-1971-0281880-6
Tibor Radó, On multiplicity functions associated with Lebesgue area, Rend. Circ. Mat. Palermo (2) 4 (1955), 219–236. MR 74498, DOI 10.1007/BF02849296