Abstract
AbstractIn this paper, we study area-filling curves, i.e. continuous and injective mappings defined on [0, 1] whose graph has positive measure. Current literature calls them “Osgood curves”, but their invention is due to H. Lebesgue. Stromberg and Tseng constructed homogeneous area-filling curves and offered an elegant example. We show that an appropriate variant of Knopp’s construction attains the same homogeneity result. In Section 4, we discuss briefly the existence of an “invasive” curve, i.e. a continuous and injective mapping from the half-open interval [0, 1[ to the unit square, whose image has measure 1. In the last section, we discuss several aspects of the Lance-Thomas curve, connecting it with the other construction due to Stromberg and Tseng.
Publisher
Springer Science and Business Media LLC
Reference22 articles.
1. Balcerzak, M., Kharazishvili, A.: On uncountable unions and intersections of measurable sets. Georgian Math. J. 6(3), 201–212 (1999)
2. Falconer, K.J.: The Geometry of Fractal Sets. Cambridge University Press, Cambridge (1985)
3. Fréchet, M.: Sur quelques points du calcul fonctionnel. Rend. Palermo 27, 1–74 (1906)
4. Gelbaum, B.R., Olmsted, J.M.H.: Counterexamples in Analysis. The Mathesis Series Holden-Day, Inc., San Francisco, Calif.-London-Amsterdam (1964)
5. Goffman, C., Pedrick, G.: A proof of the homeomorphism of Lebesgue-Stieltjes measure with Lebesgue measure. Proc. Amer. Math. Soc. 52, 196–198 (1975)
Cited by
1 articles.
订阅此论文施引文献
订阅此论文施引文献，注册后可以免费订阅5篇论文的施引文献，订阅后可以查看论文全部施引文献