Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-09T22:06:41.996Z Has data issue: false hasContentIssue false
  • English
  • Français

On measure derivation in metric spaces

Part of: Set functions and measures on spaces with additional structure

Published online by Cambridge University Press:  26 February 2010

Pio Andrea Zanzotto
Pio Andrea Zanzotto
Affiliation:
Dipartimento di Matematica dell' Universita, v. Buonarroti 2, 56100 Pisa, Italy.
Get access

Abstract

We consider a metric space (X, ρ) of a certain class studied by H. Federer in “Geometric Measure Theory”. Let Ф be any derivation basis on X, which is formed by open balls and is ρ-fine. We show that Ф allows mutual derivation of two arbitrary Borel regular measures on X, which are σ-finite and finite-valued on bounded sets. The proof is based on the so-called De Giorgi property studied in a previous paper.

MSC classification

Secondary: 28C15: Set functions and measures on topological spaces (regularity of measures, etc.)
Type
Research Article
Information
Mathematika , Volume 36 , Issue 2 , December 1989 , pp. 349 - 358
Copyright
Copyright © University College London 1989

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Brickell, F. and Clark, R. S.. Differentiable manifolds: an introduction (Van Nostrand, London, 1970).Google Scholar
2.Federer, H.. Geometric measure theory (Springer, 1969).Google Scholar
3.Giuliano-Antonini, R. and Zanzotto, P. A.. A general “geometric” condition for measure derivation. Rendiconti Accademia Nazionale delle Scienze detta dei XL, Memorie di Matematica, 105°, Vol. XI, fasc. 12 (1987), 173–191.Google Scholar
4.Hayes, C. and Pauc, C.. Derivation and Martingales (Springer, 1970).CrossRefGoogle Scholar
5.Kelley, J. L.. General Topology (Van Nostrand, New York, 1955).Google Scholar
6.Kobayashi, S. and Nomizu, K.. Foundations of differential geometry, Vol. I (Interscience, New York, 1963).Google Scholar
7.Morse, A. P.. A theory of covering and differentiation. Trans. Amer. Math. Soc, 55 (1944), 205–235.CrossRefGoogle Scholar
8.Morse, A. P.. Perfect blankets. Trans. Amer. Math. Soc, 61 (1947), 418–442.CrossRefGoogle Scholar