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On the Selection of Compact Subsets of Positive Measure from Analytic Sets of Positive Measure

Published online by Cambridge University Press:  20 November 2018

D. G. Larman
D. G. Larman*
Affiliation:
University of British Columbia, Vancouver, British Columbia; University College London, Gower St., London
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An important but seemingly difficult problem is to decide whether or not an analytic set A of positive h-measure, for some continuous Hausdorff function h, contains a compact subset C of positive h-measure, in every complete separable metric space Ω.

By extending some earlier work of R. O. Davies [1], M. Sion and D. Sjerve [8] proved that

  • (i) the selection of the set C is always possible in a σ-compact metric space Ω. More recently Davies [2] has shown that it is always possible to select C

  • (ii) when h(t) = ts, t ≧ 0, for some fixed positive number s,

  • (iii) when Ω is finite dimensional in the sense of [4],

  • (iv) when A has σ-finite h-measure, and

  • (v) when Ω is an ultra metric space.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 26 , Issue 3 , June 1974 , pp. 665 - 677
Copyright
Copyright © Canadian Mathematical Society 1974

References

1. Davies, R. O., Non-a-finite closed subsets of analytic sets, Proc. Cambridge Philos. Soc. 52 (1956), 174–7.Google Scholar
2. Davies, R. O., Increasing sequences of sets and Hausdorff measure, Proc. London Math. Soc. 20 (1970), 222–236.Google Scholar
3. Goodey, P. R., Generalized Hausdorff dimension, Mathematika 17 (1970), 324–27.Google Scholar
4. Larman, D. G., A new theory of dimension, Proc. London Math. Soc. 17 (1967), 168–192.Google Scholar
5. Larman, D. G., Projecting and uniformising Borel sets with Ka-sections, I, Mathematika 19 (1972), 231–244.Google Scholar
6. Rogers, C. A., Hausdorff measures (Cambridge University Press, Cambridge, 1970).Google Scholar
7. Sierpinski, W., Introduction to general topology (Toronto, 1934).Google Scholar
8. Sion, M. and Sjerve, D., Approximation properties of measures generated by continuous set functions, Mathematika 9 (1962), 145–56.Google Scholar