On the approximation of measurable functions by continuous functions

  • J. M. Aldaz


We present a characterization of the completed Borel measure spaces for which every measurable function, with values in a separable Frechet space, is the almost everywhere limit of a sequence of continuous functions. From this characterization one can easily obtain results that have appeared recently in the literature, in a more general form. We also examine what happens when the range is a subset of an arbitrary Banach space, and show that this case often reduces to the separable case.

1980 Mathematics Subject Classification (1985 Revision)

28C15 54D20 

Key words and phrases

τ-smooth measures Lindelöf subsets 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [AOO] Amemiya I., Okada S., Okazaki Y.,Pre-Radon measures on topological spaces, Kodai. Math. J.,1 (1978), 101–132.MATHCrossRefMathSciNetGoogle Scholar
  2. [B] Billingsley P.,Convergence of Probability Measures, Wiley, New York and London, 1968.MATHGoogle Scholar
  3. [BP] Bessaga C., Pelczyńsky A.,Infinite-dimensional Topology, Polish Scientific Publishers, 1975.Google Scholar
  4. [DU] Diestel J., Uhl J. J., Jr.,Vector Measures, Mathematical Surveys, 15, Amer. Math. Soc., 1977.Google Scholar
  5. [E] Edgar G. A.,Measurability in a Banach space, Indiana Univ. Math.,26 no. 4 (1977), 663–677.MATHCrossRefMathSciNetGoogle Scholar
  6. [En] Engelking R.,General Topology, Heldermann Verlag Berlin, 1989.MATHGoogle Scholar
  7. [F] Fremlin D. H.,Measurable functions and almost continuous functions, Manuscripta. Math.,33 (1981), 387–405.MATHCrossRefMathSciNetGoogle Scholar
  8. [GP] Gardner R. J., Pfeffer W. F.,Borel Measures, Handbook of settheoretic topology (K. Kunen and J. E. Vaughan, eds.), North-Holland, 1984, pp. 961–1043.Google Scholar
  9. [H] Halmos P. R.,Measure Theory, Van Nostrand, 1950.Google Scholar
  10. [K] Koumoullis G.,On perfect measures, Trans. Amer. Math. Soc.,264 no. 2 (1981), 521–537.MATHCrossRefMathSciNetGoogle Scholar
  11. [KP] Kupka J., Prikry K.,The measurability of uncountable unions, Amer. Math. Monthly,91 no. 2 (1984), 85–97.MATHCrossRefMathSciNetGoogle Scholar
  12. [M] Munkres J. R.,Topology, a first course, Prentice-Hall, 1975.Google Scholar
  13. [MS] Marczewski E., Sikorski R.,Measures on non-seprable metric spaces, Colloq. Math.,1 (1948), 133–149.MATHMathSciNetGoogle Scholar
  14. [W] Wheeler R. F.,A survey of Baire measures and strict topologies, Expo. Math.,77 (1983), 97–190.MathSciNetGoogle Scholar
  15. [Wi] Wisniewski A.,The structure of measurable mappings on metric spaces, Proc. Amer. Math. Soc.,122 no. 1 (1994), 147–150.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer 1996

Authors and Affiliations

  1. 1.Departmento de Matemáticas Faciltad de CienciasUniversidad Autónoma de MadridMadridSpain

Personalised recommendations