Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-22T16:14:35.269Z Has data issue: false hasContentIssue false
  • English
  • Français

Strictly positive measures on Boolean algebras

Published online by Cambridge University Press:  12 March 2014

Mirna Džamonja  and
Grzegorz Plebanek
Mirna Džamonja
Affiliation:
School of Mathematics, University of East Anglia, Norwich NR47TJ, UK, E-mail: [email protected]
Grzegorz Plebanek
Affiliation:
Institute of Mathematics, Wrocław University, 2/4 PL. Grunwaldzki, 50-384 Wrocław, Poland, E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We investigate strictly positive finitely additive measures on Boolean algebras and strictly positive Radon measures on compact zerodimensional spaces. The motivation is to find a combinatorial characterisation of Boolean algebras which carry a strictly positive finitely additive finite measure with some additional properties, such as separability or nonatomicity. A possible consistent characterisation for an algebra to carry a separable strictly positive measure was suggested by Talagrand in 1980, which is that the Stone space K of the algebra satisfies that its space M(K) of measures is weakly separable, equivalently that C(K) embeds into l∞. We show that there is a ZFC example of a Boolean algebra (so of a compact space) which satisfies this condition and does not support a separable strictly positive measure. However, we use this property as a tool in a proof which shows that under MA + ¬ CH every atomless ccc Boolean algebra of size < c carries a nonatomic strictly positive measure. Examples are given to show that this result does not hold in ZFC. Finally, we obtain a characterisation of Boolean algebras that carry a strictly positive nonatomic measure in terms of a chain condition, and we draw the conclusion that under MA + ¬ CH every atomless ccc Boolean algebra satisfies this stronger chain condition.

Keywords

strictly positive measurechain conditions
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 73 , Issue 4 , December 2008 , pp. 1416 - 1432
Copyright
Copyright © Association for Symbolic Logic 2008

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

REFERENCES

[1]Argyros, S. A., On compact spaces without strictly positive measure, Pacific Journal of Mathematics, vol. 105 (1983). no. 2, pp. 257–272.CrossRefGoogle Scholar
[2]Argyros, S. A., Mercourakis, S., and Negrepontis, S., Functional-analytic properties of Corson-compact space, Studia Mathematica, vol. 89 (1988), no. 3, pp. 197–229.CrossRefGoogle Scholar
[3]Balcar, B., Jech, T., and Pazák, T., Complete ccc boolean algebras, the order sequential topology, and a problem of von Neumann, Bulletin of the London Mathematical Society, vol. 37 (2005). no. 6, pp. 885–898.CrossRefGoogle Scholar
[4]Blaszczyk, A. and Szymański, A., Cohen algebras and nowhere dense ultrafilters, Bulletin of the Polish Academy of Sciences, vol. 49 (2001), no. 1, pp. 15–25.Google Scholar
[5]Ciesielski, K. and Pawlikowski, J., The covering property axiom, CPA, Cambridge University Press, 2004.CrossRefGoogle Scholar
[6]Comfort, W. W. and Negrepontis, S., Chain conditions in topology, Cambridge University Press, 1982.CrossRefGoogle Scholar
[7]Dow, A. and Steprans, J., The σ-linkedness of measure algebra, Canadian Mathematical Bulletin, vol. 37 (1993). no. 1, pp. 42–45.CrossRefGoogle Scholar
[8]Engelking, R., General topology, 2nd ed., Sigma Series in Pure Mathematics, vol. 6, Heldermann Verlag, Berlin, 1980. Translated from the Polish by the author.Google Scholar
[9]Farah, I., Examples of epsilon-exhaustive submeasures, Fundamenta Mathematicae, vol. 181 (2004), pp. 257–272.CrossRefGoogle Scholar
[10]Farah, I. and Veličković, B., von Neumann's problem and large cardinals, Bulletin of the London Mathematical Society, vol. 38 (2006), no. 6, pp. 907–912.CrossRefGoogle Scholar
[11]Fremlin, D. H., Consequences of Martin's axiom, Cambridge University Press, 1984.CrossRefGoogle Scholar
[12]Fremlin, D. H., Measure algebras, Handbook of Boolean algebras (Monk, J. D., editor), vol. III, North–Holland, 1989.Google Scholar
[13]Fremlin, D. H., Measure theory, Measure algebras, vol. 3, Torres Fremlin, 2002.Google Scholar
[14]Fremlin, D. H., Measure theory, Topological measure spaces, vol. 4, Torres Fremlin, 2003.Google Scholar
[15]Gaifman, H., Concerning measures on Boolean algebras, Pacific Journal of Mathematics, vol. 14 (1964). pp. 61–73.CrossRefGoogle Scholar
[16]Gardner, R. J. and Pfeffer, W. F., Borel measures, Handbook of set-theoretic topology (Kunen, K. and Vaughan, J. E., editors), Elsevier Science Publishers B. V., 1984, pp. 961–1043.CrossRefGoogle Scholar
[17]Horn, A. and Tarski, A., Measures in Boolean algebras, Transactions of the American Mathematical Society, vol. 64 (1948), no. 3, pp. 467–497.CrossRefGoogle Scholar
[18]Juhász, I. and Szymanski, A., d-calibers and d-tightness in compact spaces, Topology and its Applications, vol. 151 (2005), pp. 66–76.CrossRefGoogle Scholar
[19]Kelley, J. L., Measures on Boolean algebras, Pacific Journal of Mathematics, vol. 9 (1959), pp. 1165–1177.CrossRefGoogle Scholar
[20]Mägerl, G. and Namioka, I., Intersection numbers and weak* separability of spaces of measures, Mathematische Annalen, vol. 249 (1980), no. 3, pp. 273–279.CrossRefGoogle Scholar
[21]Maharam, D., Problem 167, The Scottish book (Mauldin, D., editor), Birhäuser, 1981, pp. 240–243.Google Scholar
[22]Sikorski, R., Boolean algebras, Springer-Verlag, 1964.Google Scholar
[23]Simon, P., A countable dense-in-itself dense P-set, Topology and Its Applications, vol. 123 (2002), pp. 193–198.CrossRefGoogle Scholar
[24]Strassen, V., The existence of probability measures with given marginals, Annals of Mathematical Statistics, vol. 36 (1965), pp. 423–439.CrossRefGoogle Scholar
[25]Talagrand, M., Séparabilité vague dans l'espace des mesures sur un compact, Israel Journal of Mathematics, vol. 37 (1980), no. 1–2, pp. 171–180.CrossRefGoogle Scholar
[26]Talagrand, M., Maharam's problem, Annals of Mathematics, to appear.Google Scholar
[27]Todorčević, S., Chain-condition methods in topology, Topology and Its Applications, vol. 101 (2000), pp. 45–82.CrossRefGoogle Scholar
[28]Todorčević, S., A problem of von Neumann and Maharam about algebras supporting continuos submeasures, Fundamenta Mathematicae, vol. 183 (2004), no. 2, pp. 169–183.CrossRefGoogle Scholar
[29]Veličković, B., CCC forcings and splitting reals, Israel Journal of Mathematics, vol. 147 (2005), pp. 209–220.CrossRefGoogle Scholar