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Invariant Daniell integrals

Abstract Integration

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Part of the Lecture Notes in Mathematics book series (LNM,volume 1089)

Abstract

Let X be a Hausdorff space and let L be a sublattice of the vector lattice of real continuous functions on X. Consider a localizable Daniell integral μ on L, i.e. one which is defined by a Radon measure m on X by the formula μ(φ)=∫φdm. Then if G is a group of homeomorphisms of X leaving L invariant, it is shown that under appropriate hypotheses, the invariance of μ under the action of G implies the quasi-invariance of a certain class of measures associated with m on a quotient space Y of X. Conversely, if G is locally compact and Y=G/H, the class of quasi-invariant Radon measures on Y is associated in this way to a G-invariant Daniell integral on a certain half-line bundle over Y.

Keywords

  • Radon Measure
  • Hausdorff Space
  • Measurable Selection
  • Continuous Section
  • Invariant Section

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References

  1. N. Bourbaki, Intégration ch.7, Hermann.

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  2. E.G.F. Thomas and A. Volčič, Daniell integrals represented by Radon measures, in preparation.

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  3. E.G.F. Thomas, Integral representation in Convex cones. Report, University of Groningen ZW 7703, 1977.

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© 1984 Springer-Verlag

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Thomas, E.G.F. (1984). Invariant Daniell integrals. In: Kölzow, D., Maharam-Stone, D. (eds) Measure Theory Oberwolfach 1983. Lecture Notes in Mathematics, vol 1089. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0072610

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  • DOI: https://doi.org/10.1007/BFb0072610

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-13874-7

  • Online ISBN: 978-3-540-39069-5

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