Additivity of quasi-measures
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- by D. J. Grubb and Tim LaBerge PDF
- Proc. Amer. Math. Soc. 126 (1998), 3007-3012 Request permission
Abstract:
We prove that quasi-measures on compact Hausdorff spaces are countably additive. Contained in this result is a proof that every quasi-measure decomposes uniquely into a measure and a quasi-measure that has no smaller measure beneath it. We also show that it is consistent with the usual axioms of set-theory that quasi-measures on compact Hausdorff spaces are $\aleph _1$-additive. Finally, we construct an example that places strong restrictions on other forms of additivity.References
- Johan F. Aarnes, Quasi-states and quasi-measures, Adv. Math. 86 (1991), no. 1, 41–67. MR 1097027, DOI 10.1016/0001-8708(91)90035-6
- Johan F. Aarnes, Pure quasi-states and extremal quasi-measures, Math. Ann. 295 (1993), no. 4, 575–588. MR 1214949, DOI 10.1007/BF01444904
- Johan F. Aarnes, Construction of non-subadditive measures and discretization of Borel measures, Fund. Math. 147 (1995), no. 3, 213–237. MR 1348720, DOI 10.4064/fm-147-3-213-237
- J. Boardman. Quasi-measures on completely regular spaces, Rocky Mountain J. Math., To appear.
- Ryszard Engelking, General topology, 2nd ed., Sigma Series in Pure Mathematics, vol. 6, Heldermann Verlag, Berlin, 1989. Translated from the Polish by the author. MR 1039321
- D. H. Fremlin, Consequences of Martin’s axiom, Cambridge Tracts in Mathematics, vol. 84, Cambridge University Press, Cambridge, 1984. MR 780933, DOI 10.1017/CBO9780511896972
- Gary Gruenhage, Partitions of compact Hausdorff spaces, Fund. Math. 142 (1993), no. 1, 89–100. MR 1207473, DOI 10.4064/fm-142-1-89-100
- Kenneth Kunen, Set theory, Studies in Logic and the Foundations of Mathematics, vol. 102, North-Holland Publishing Co., Amsterdam-New York, 1980. An introduction to independence proofs. MR 597342
- W. Sierpinski. Un théorème sur les continus, Tôhoku Math. J. 13 (1918) 300–303.
- Robert F. Wheeler, Quasi-measures and dimension theory, Topology Appl. 66 (1995), no. 1, 75–92. MR 1357876, DOI 10.1016/0166-8641(95)00009-6
Additional Information
- D. J. Grubb
- Affiliation: Department of Mathematical Sciences, Northern Illinois University, Dekalb, Illinois 60115
- Email: grubb@math.niu.edu
- Tim LaBerge
- Affiliation: Department of Mathematical Sciences, Northern Illinois University, Dekalb, Illinois 60115
- Email: laberget@math.niu.edu
- Received by editor(s): December 23, 1996
- Received by editor(s) in revised form: March 13, 1997
- Communicated by: Dale E. Alspach
- © Copyright 1998 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 126 (1998), 3007-3012
- MSC (1991): Primary 28C15
- DOI: https://doi.org/10.1090/S0002-9939-98-04494-3
- MathSciNet review: 1458874