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)



Regular and purely irregular bounded charges: a decomposition theorem

Authors: Bruno Girotto and Silvano Holzer
Journal: Proc. Amer. Math. Soc. 116 (1992), 683-693
MSC: Primary 28C15; Secondary 60B05
MathSciNet review: 1099340
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We introduce the notions of regular and purely irregular charges with respect to a pair of pavings and study their structural properties. Moreover, we link regularity and $ \sigma $-additivity, obtaining some generalizations of well-known theorems. Finally, when the pavings satisfy some reasonable weak conditions, we can decompose any bounded charge into regular and purely irregular decomposants; this decomposition becomes the Hewitt-Yosida one, whenever the charges are defined on the Baire $ \sigma $-field of a countably compact space.

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

  • [1] A. D. Alexandroff, Additive set-functions in abstract spaces, Rec. Math. [Mat. Sbornik] N.S. 9 (51) (1941), 563–628 (English, with Russian summary). MR 0005785
  • [2] Thomas E. Armstrong, Strong singularity, disjointness, and strong finite additivity of finitely additive measures, J. Math. Anal. Appl. 131 (1988), no. 2, 565–587. MR 935289, 10.1016/0022-247X(88)90226-0
  • [3] Thomas E. Armstrong and Karel Prikry, Singularity and absolute continuity with respect to strategic measures, Illinois J. Math. 27 (1983), no. 4, 624–658. MR 720099
  • [4] K. P. S. Bhaskara Rao and M. Bhaskara Rao, Theory of charges, Pure and Applied Mathematics, vol. 109, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983. A study of finitely additive measures; With a foreword by D. M. Stone. MR 751777
  • [5] S. Bochner and R. S. Phillips, Additive set functions and vector lattices, Ann. of Math. (2) 42 (1941), 316–324. MR 0004079
  • [6] R. J. Gardner, The regularity of Borel measures, Measure theory, Oberwolfach 1981 (Oberwolfach, 1981) Lecture Notes in Math., vol. 945, Springer, Berlin-New York, 1982, pp. 42–100. MR 675272
  • [7] Konrad Jacobs, Measure and integral, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. Probability and Mathematical Statistics; With an appendix by Jaroslav Kurzweil. MR 514702
  • [8] Washek F. Pfeffer, Integrals and measures, Marcel Dekker, Inc., New York-Basel, 1977. Monographs and Textbooks in Pure and Applied Mathematics, Vol. 42. MR 0460580
  • [9] Flemming Topsøe, Topology and measure, Lecture Notes in Mathematics, Vol. 133, Springer-Verlag, Berlin-New York, 1970. MR 0422560

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 28C15, 60B05

Retrieve articles in all journals with MSC: 28C15, 60B05

Additional Information

Keywords: Normal subspace decomposition, normal decomposition of a bounded charge, paving, regular and purely irregular charges, $ \sigma $-compact set system, Baire and Borel sets
Article copyright: © Copyright 1992 American Mathematical Society