Abstract
If Cāāā2ā is the Cantor set realized as the infinite product of two-point groups, then a folklore result says the Cantor map from C into [0,1] sends Haar measure to Lebesgue measure on the interval. In fact, C admits many distinct topological group structures. In this note, we show that the Haar measures induced by these distinct group structures are all the same. We prove this by showing that Haar measure for any group structure is the same as Haar measure induced by a related abelian group structure. Moreover, each abelian group structure on C supports a natural total order that determines a map onto the unit interval that is monotone, and hence sends intervals in C to subintervals of the unit interval. Using techniques from domain theory, we show this implies this map sends Haar measure on C to Lebesgue measure on the interval, and we then use this to prove any two group structures on C have the same Haar measure.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Abramsky, S.: Domain theory in logical form. Annals of Pure and Applied LogicĀ 51, 1ā77 (1991)
Abramsky, S., Jung, A.: Domain Theory. In: Handbook of Logic in Computer Science, pp. 1ā168. Clarendon Press (1994)
Fedorchuk, V.: Probability measures in topology. Russ. Math. Surv.Ā 46, 45ā93 (1991)
Gierz, G., Hofmann, K.H., Lawson, J.D., Mislove, M., Scott, D.: Continuous Lattices and Domains. Cambridge University Press (2003)
Gehrke, M., Grigorieff, S., Pin, J.-Ć.: Duality and equational theory of regular languages. In: Aceto, L., DamgĆ„rd, I., Goldberg, L.A., HalldĆ³rsson, M.M., IngĆ³lfsdĆ³ttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part II. LNCS, vol.Ā 5126, pp. 246ā257. Springer, Heidelberg (2008)
Gehrke, M.: Stone duality and the recognisable languages over an algebra. In: Kurz, A., Lenisa, M., Tarlecki, A. (eds.) CALCO 2009. LNCS, vol.Ā 5728, pp. 236ā250. Springer, Heidelberg (2009)
Hofmann, K.H., Mislove, M.: Compact affine monoids, harmonic analysis and information theory, in: Mathematical Foundations of Information Flow. AMS Symposia on Applied MathematicsĀ 71, 125ā182 (2012)
Hofmann, K.H., Morris, S.: The Structure Theory of Compact Groups, de Gruyter Studies in Mathematics, 2nd edn., vol.Ā 25, p. 858. de Gruyter Publishers (2008)
Jones, C.: Probabilistic Nondeterminism, PhD Thesis, University of Edinburgh (1988)
Jung, A., Tix, R.: The troublesome probabilistic powerdomain. ENTCSĀ 13, 70ā91 (1998)
Mislove, M.: Topology. domain theory and theoretical computer science. Topology and Its ApplicationsĀ 89, 3ā59 (1998)
Rotman, J.: An Introduction to the Theory of Groups, Graduate Studies in Mathematics, 4th edn. Springer (1999)
Saheb-Djarhomi, N.: CPOs of measures for nondeterminism. Theoretical Computer ScienceĀ 12, 19ā37 (1980)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
Ā© 2014 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Brian, W., Mislove, M. (2014). From Haar to Lebesgue via Domain Theory. In: van Breugel, F., Kashefi, E., Palamidessi, C., Rutten, J. (eds) Horizons of the Mind. A Tribute to Prakash Panangaden. Lecture Notes in Computer Science, vol 8464. Springer, Cham. https://doi.org/10.1007/978-3-319-06880-0_11
Download citation
DOI: https://doi.org/10.1007/978-3-319-06880-0_11
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-06879-4
Online ISBN: 978-3-319-06880-0
eBook Packages: Computer ScienceComputer Science (R0)