Skip to main content
SpringerLink
Log in

Measures

  • Chapter
  • First Online:
Banach Spaces of Continuous Functions as Dual Spaces

Part of the book series: CMS Books in Mathematics ((CMSBM))

  • 1189 Accesses

Abstract

In this chapter, we shall study the (complex) Banach lattice M(K) consisting of all complex-valued, regular Borel measures on a locally compact space K and, in particular, the positive measures in M(K), which form the cone M(K)+. The Banach space M(K) is isometrically isomorphic to the dual of C  0(K). In \(\S\)4.2, we shall discuss the linear spaces of discrete measures and of continuous measures on K.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 79.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 99.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 129.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. C. A. Akemann, The dual space of an operator algebra. Trans. American Math. Soc., 126, 286–302 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  2. F. Albiac, N. J. Kalton, Topics in Banach Space Theory. Graduate Texts in Mathematics, vol. 233 (Springer, New York, 2006)

    Google Scholar 

  3. E. M. Alfsen, Compact Convex Sets and Boundary Integtrals. Ergebnisse der Mathematik und ihrer Granzgebiete, Band 57 (Springer, New York, 1971)

    Google Scholar 

  4. W. G. Bade, The Space of all Continuous Functions on a Compact Hausdorff Space (University of California, Berkeley, 1957). Library call no. QA689.B16

    Google Scholar 

  5. W. G. Bade, The Banach Space C(S). Lecture Note Series, vol. 26 (Matematisk Institut, Aarhus Universitët, Aarhus, 1971)

    Google Scholar 

  6. S. Banach, Théorie des Opérations Linéaires. Monografie Matematyczne, vol. 1 (Instytut Matematyczny Polskiej Akademii Nauk, Warsaw, 1932)

    Google Scholar 

  7. G. Birkhoff, Lattice Theory, 3rd edn. (American Mathematical Society, Providence, 1967)

    MATH  Google Scholar 

  8. V. I. Bogachev, Measure Theory, vols. I, II (Springer, Berlin, 2007)

    Book  MATH  Google Scholar 

  9. T. Budak, N. Işik, J. Pym, Minimal determinants of topological centres for some algebras associated with locally compact groups. Bull. London Math. Soc. 43, 495–506 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. C. Caratheodory, Die homomorphieen von somen und die multiplikation von inhaltsfunktionen. Annalen Scuola Norm. Sup. Pisa Cl. Sci. 8 (2), 105–130 (1939)

    MathSciNet  MATH  Google Scholar 

  11. D. L. Cohn, Measure Theory, 2nd edn. (Birkhäuser/Springer, New York, 2013)

    Book  MATH  Google Scholar 

  12. W. W. Comfort, Topological groups, in Handbook of Set Theoretic Topology, ed. by K. Kunen, J.E. Vaughan (North Holland, New York, 1984), pp. 1143–1263

    Chapter  Google Scholar 

  13. H. G. Dales, Banach Algebras and Automatic Continuity. London Mathematical Society Monographs, vol. 24 (Clarendon Press, Oxford, 2000)

    Google Scholar 

  14. H. G. Dales, A. T.-M. Lau, D. Strauss, Banach algebras on semigroups and on their compactifications. Mem. American Math. Soc. 205, 165 (2010)

    MathSciNet  MATH  Google Scholar 

  15. H. G. Dales, A. T.-M. Lau, D. Strauss, Second duals of measure algebras. Diss. Math. (Rozprawy Matematyczne) 481, 121 (2012)

    Google Scholar 

  16. F. K. Dashiell Jr., Nonweakly compact operators from order-Cauchy complete C(S) lattices, with application to Baire classes. Trans. American Math. Soc. 266, 397–413 (1981)

    MathSciNet  MATH  Google Scholar 

  17. J. Diestel, J. J. Uhl Jr., Vector Measures. Mathematical Surveys, vol. 15 (American Mathematical Society, Providence, 1977)

    Google Scholar 

  18. J. Dixmier, Sur certains espaces considérés par M. H. Stone. Summa Brasiliensis Math. 2, 151–182 (1951)

    MathSciNet  MATH  Google Scholar 

  19. N. Dunford, J. T. Schwartz, Linear Operators, Part I: General Theory (Interscience Publishers, New York, 1957)

    MATH  Google Scholar 

  20. F. J. Fernández-Polo, A. M. Peralta, A short proof of a theorem of Pfitzner. Quarterly Journal Math. 61, 329–336 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  21. B. Fishel, D. Papert, A note on hyperdiffuse measures. J. London Math. Soc. 39, 245–254 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  22. V. P. Fonf, J. Lindenstrauss, R. R. Phelps, Infinite dimensional convexity, in Handbook of the Geometry of Banach Spaces, vol. 1, ed. by W. B. Johnson, J. Lindenstrauss (North Holland/Elsevier, Amsterdam, 2001), pp. 599–670

    Chapter  Google Scholar 

  23. D. H. Fremlin, Measure Theory, vol. 3, 2012 edn. This text is available at www.essex.ac.uk/maths/people/fremlin/mt.htm

  24. D. H. Fremlin, G. Plebanek, Large families of mutually singular Radon measures. Bull. Pol. Acad. Sci. Math. 51, 169–174 (2003)

    MathSciNet  MATH  Google Scholar 

  25. I. M. Gelfand, Abstrakte Functionen und Lineare Operatoren. Recueil Math. 4, 235–286 (1938)

    MATH  Google Scholar 

  26. C. Goffman, Real Functions (Prindle/Weber/Schmidt, Boston, 1953)

    MATH  Google Scholar 

  27. A. Grothendieck, Sur les appplications lineaires faiblement compactes d’espaces du type C(K). Canadian J. Math. 5, 129–173 (1953)

    Article  MathSciNet  MATH  Google Scholar 

  28. P. R. Halmos, Measure Theory (D. van Nostrand, New York, 1950)

    Book  MATH  Google Scholar 

  29. E. Hewitt, K. A. Ross, Abstract Harmonic Analysis I, 2nd edn. (Springer, Berlin, 1979)

    MATH  Google Scholar 

  30. E. Hewitt, K. Stromberg, Real and Abstract Analysis (Springer, New York, 1975)

    MATH  Google Scholar 

  31. A. Kolmogorov, General measure theory and the calculus of probabilities. Trudy Kommunist Akad. Razd. mat. 8–21 (1929, in Russian). English transl.: Selected works of A. N. Kolmogorov, vol. II, ed. by A. N. Shiryayev (Kluwer Academic Publishers, Dordrecht, 1992), pp. 48–59

    Google Scholar 

  32. H. E. Lacey, Isometric Theory of Classical Banach Spaces (Springer, Berlin, 1974)

    Book  MATH  Google Scholar 

  33. A. T.-M. Lau, V. Losert, Complementation of certain subspaces of L ∞ (G) of a locally compact group. Pacific J. Math. 141, 295–310 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  34. D. R. Lewis, C. Stegall, Banach spaces whose duals are isomorphic to ℓ 1(Γ). J. Functional Anal. 12, 177–187 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  35. J. Lukeš, J. Malý, I. Netuka, J. Spurný, Integral Representation Theory (Walter de Gruyter, Berlin, 2010)

    MATH  Google Scholar 

  36. W. A. J. Luxemburg, Is every integral normal? Bull. American Math. Soc. 73, 685–688 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  37. W. A. J. Luxemburg, A. C. Zaanan, Riesz Spaces, vol. I (North Holland, Amsterdam, 1971)

    Google Scholar 

  38. D. Maharam, Decompositions of measure algebras. Proc. Natl. Acad. Sci. USA 28, 142–160 (1942)

    Article  MATH  Google Scholar 

  39. P. Meyer-Nieberg, Banach Lattices (Springer, Berlin, 1991)

    Book  MATH  Google Scholar 

  40. T. W. Palmer, Banach Algebras and the General Theory of ∗-Algebras, vol. I, Algebras and Banach Algebras (Cambridge University Press, Cambridge, 1994)

    Google Scholar 

  41. T. W.Ṗalmer, Banach Algebras and the General Theory of ∗-Algebras, vol. II, ∗-Algebras (Cambridge University Press, Cambridge, 2001)

    Google Scholar 

  42. A. Pełczyński, On the isomorphism of the spaces m and M. Bull. Acad. Pol. Sci. 6, 695–696 (1958)

    Google Scholar 

  43. A. Pełczyński, On Banach spaces containing L 1(μ). Studia Math. 19, 231–246 (1968)

    MathSciNet  MATH  Google Scholar 

  44. A. Pełczyński, Linear extensions, linear averagings, and their applications to linear topological classification of spaces of continuous functions. Diss. Math. (Rozprawy Matematyczne) 63, 92 (1968)

    Google Scholar 

  45. H. Pfitzner, Weak compactness in the dual of a C ∗-algebra is determined commutatively. Math. Annalen 298, 349–371 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  46. R. Phelps, Lectures on Choquet’s Theorem. van Nostrand Mathematical Studies, vol. 7 (van Nostrand, Princeton, 1966). Second edition, Lecture Notes in Mathematics, vol. 1757 (Springer, Berlin, 2001)

    Google Scholar 

  47. R. S. Phillips, On linear transformations. Trans. American Math. Soc. 48, 516–541 (1940)

    Article  MATH  Google Scholar 

  48. G. Plebanek, A normal measure on a compact connected space. Unpublished notes (2014)

    Google Scholar 

  49. H. P. Rosenthal, On injective Banach spaces and the spaces L ∞(μ) for finite measures μ. Acta Math. 124, 205–248 (1970)

    Google Scholar 

  50. H. L. Royden, Real Analysis, 3rd edn. (Prentice-Hall, Engelwood Cliffs, 1988)

    MATH  Google Scholar 

  51. W. Rudin, Real and Complex Analysis, 3rd edn. (McGraw-Hill, New York, 1986)

    MATH  Google Scholar 

  52. W. Rudin, Functional Analysis, 2nd edn. (McGraw-Hill, New York, 1991)

    MATH  Google Scholar 

  53. K. Saitô, J. D. M. Wright, C ∗-algebras which are Grothendieck spaces. Rend. Circ. Mat. Palermo, Serie II 52, 141–144 (2003)

    Google Scholar 

  54. K. Saitô, J. D. M. Wright, Monotone Complete C ∗ -Algebras and Generic Dynamics. Springer Monographs in Mathematics (Springer, London, 2015)

    Google Scholar 

  55. G. L. Seever, Measures on F-spaces. Trans. American Math. Soc. 133, 267–280 (1968)

    MathSciNet  MATH  Google Scholar 

  56. Z. Semadeni, Banach Spaces of Continuous Functions. Monografie Matematyczne, vol. 55 (Instytut Matematyczny Polskiej Akademii Nauk, Warsaw, 1971)

    Google Scholar 

  57. E. Szpilrajn, On the space of measurable sets. Annalen Soc. Pol. Math. 17, 120–121 (1938)

    Google Scholar 

  58. M. Takesaki, Theory of Operator Algebras I (Springer, New York, 1979)

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Lancaster, Lancaster, UK

    H. G. Dales

  2. Center of Excellence in Computation Algebra, and Topology (CECAT), Chapman University, Orange, CA, USA

    F. K. Dashiell Jr.

  3. Department of Mathematical Sciences, University of Alberta, Edmonton, AB, Canada

    A. T.-M. Lau

  4. Department of Pure Mathematics, University of Leeds, Leeds, UK

    D. Strauss

Authors
  1. H. G. Dales
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. F. K. Dashiell Jr.
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. T.-M. Lau
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. D. Strauss
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Dales, H.G., Dashiell, F.K., Lau, A.TM., Strauss, D. (2016). Measures. In: Banach Spaces of Continuous Functions as Dual Spaces. CMS Books in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-32349-7_4

Download citation

Publish with us

Policies and ethics

Search

Navigation