Current valued measures and Geöcze area

Gariepy, Ronald

doi:10.1090/S0002-9947-1972-0293066-0

Current valued measures and Geöcze area
HTML articles powered by AMS MathViewer

by Ronald Gariepy PDF

Trans. Amer. Math. Soc. 166 (1972), 133-146 Request permission

Abstract:

If f is a continuous mapping of finite Geöcze area from a polyhedral region $X \subset {R^k}$ into ${R^n},2 \leqq k \leqq n$, then, under suitable hypotheses, one can associate with f, by means of the Cesari-Weierstrass integral, a current valued measure T over the middle space of f. In particular, if either $k = 2$ or the $k + 1$-dimensional Hausdorff measure of $f(X)$ is zero, then T is essentially the same as a current valued measure defined by H. Federer and hence serves to describe the tangential properties of f and the multiplicities with which f assumes its values. Further, the total variation of T is equal to the Geöcze area of f.

References

Lamberto Cesari, Surface area, Annals of Mathematics Studies, No. 35, Princeton University Press, Princeton, N. J., 1956. MR 0074500
Lamberto Cesari, Quasi-additive set functions and the concept of integral over a variety, Trans. Amer. Math. Soc. 102 (1962), 94–113. MR 142723, DOI 10.1090/S0002-9947-1962-0142723-9
Lamberto Cesari, Extension problem for quasi-additive set functions and Radon-Nikodym derivatives, Trans. Amer. Math. Soc. 102 (1962), 114–146. MR 142724, DOI 10.1090/S0002-9947-1962-0142724-0
Maurice R. Demers and Herbert Federer, On Lebesgue area. II, Trans. Amer. Math. Soc. 90 (1959), 499–522. MR 102586, DOI 10.1090/S0002-9947-1959-0102586-4
Herbert Federer, Measure and area, Bull. Amer. Math. Soc. 58 (1952), 306–378. MR 49289, DOI 10.1090/S0002-9904-1952-09586-0
Herbert Federer, On Lebesgue area, Ann. of Math. (2) 61 (1955), 289–353. MR 67178, DOI 10.2307/1969917
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
Herbert Federer and Wendell H. Fleming, Normal and integral currents, Ann. of Math. (2) 72 (1960), 458–520. MR 123260, DOI 10.2307/1970227
J. H. Michael, The convergence of measures on parametric surfaces, Trans. Amer. Math. Soc. 107 (1963), 140–152. MR 146352, DOI 10.1090/S0002-9947-1963-0146352-3
Togo Nishiura, The Geöcze $k$-area and a cylindrical property, Proc. Amer. Math. Soc. 12 (1961), 795–800. MR 125941, DOI 10.1090/S0002-9939-1961-0125941-X
Togo Nishiura, The Geöcze $k$-area and flat mappings, Rend. Circ. Mat. Palermo (2) 11 (1962), 105–125. MR 148855, DOI 10.1007/BF02849429
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