Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Homeomorphic measures in metric spaces


Author: John C. Oxtoby
Journal: Proc. Amer. Math. Soc. 24 (1970), 419-423
MSC: Primary 28.13
DOI: https://doi.org/10.1090/S0002-9939-1970-0260961-1
MathSciNet review: 0260961
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For any nonatomic, normalized Borel measure $ \mu $ in a complete separable metric space $ X$ there exists a homeomorphism $ h:\mathfrak{N} \to X$ such that $ \mu = \lambda {h^{ - 1}}$ on the domain of $ \mu $, where $ \mathfrak{N}$ is the set of irrational numbers in $ (0,1)$ and $ \lambda $ denotes Lebesgue-Borel measure in $ \mathfrak{N}$. A Borel measure in $ \mathfrak{N}$ is topologically equivalent to $ \lambda $ if and only if it is nonatomic, normalized, and positive for relatively open subsets.


References [Enhancements On Off] (What's this?)

  • [1] P. Alexandroff and P. Urysohn, Über nulldimensionale Punktmengen, Math. Ann. 98 (1928), 89-106. MR 1512393
  • [2] N. Bourbaki, Intégration. Livre VI. Chapitre 5: Intégration des mesures, Hermann, Paris, 1956. MR 18, 881.
  • [3] B. R. Gelbaum, Cantor sets in metric measure spaces, Proc. Amer. Math. Soc. 24 (1970), 341-343. MR 0254201 (40:7411)
  • [4] L. Gillman and M. Jerison, Rings of continuous functions, The University Series of Higher Math., Van Nostrand, Princeton, N. J., 1960. MR 22 #6994. MR 0116199 (22:6994)
  • [5] P. R. Halmos, Lectures on Boolean algebras, Van Nostrand Math. Studies, no. 1, Van Nostrand, Princeton, N. J., 1963. MR 29 #4713. MR 0167440 (29:4713)
  • [6] C. Kuratowski, Topologie. Vol. I, 4 ème éd., PWN, Warsaw, 1958. MR 19, 873. MR 0090795 (19:873d)
  • [7] E. Marczewski and R. Sikorski, Measures in non-separable metric spaces, Colloq. Math. 1 (1948), 133-139. MR 10, 23. MR 0025548 (10:23f)
  • [8] J. C. Oxtoby and S. M. Ulam, Measure-preserving homeomorphisms and metrical transitivity, Ann. of Math. (2) 42 (1941), 874-920. MR 3, 211. MR 0005803 (3:211b)
  • [9] J. von Neumann, Einige Sätze über Messbare Abbildungen, Ann. of Math. (2) 33 (1932), 574-586. MR 1503077

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 28.13

Retrieve articles in all journals with MSC: 28.13


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1970-0260961-1
Keywords: Topologically equivalent Borel measures, homeomorphic measure spaces, measure-preserving mapping, complete separable metric space, space of irrational numbers, Cantor set
Article copyright: © Copyright 1970 American Mathematical Society

American Mathematical Society