Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-15T08:26:48.833Z Has data issue: false hasContentIssue false
  • English
  • Français

A Local Property of Measurable Sets

Published online by Cambridge University Press:  20 November 2018

W. Eames
W. Eames*
Affiliation:
Queen's University and Sir John Cass College
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let Ω be a metric space with metric ρ, let C be a class of closed sets from Ω and let τ be a non-negative real-valued set function on C. We assume that the empty set ϕ is in C and that τ(I)= 0 if and only if I = ϕ. For each set A in Ω, we define φ(A), 0 ≤ φ(A) ≤ ∞ by:

where the infimum is taken for all possible countable collections of sets I(n) from C such that:

and the diameter of I(n), d(I(n)), is less than ∈ for every n.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 12 , 1960 , pp. 632 - 640
Copyright
Copyright © Canadian Mathematical Society 1960

References

1. Besicovitch, A.S., On the fundamental geometrical properties of linearly measurable plane sets of points, Math. Annalen, 98 (1927), 422–464.Google Scholar
2. Jeffery, R.L., Sets of k-extent in n-dimensional space, Trans. Amer. Math. Soc, 35 (1933), 629–647.Google Scholar
3. Munroe, M.E., Introduction to measure and integration (Cambridge, Mass., 1953).Google Scholar
4. Nikodym, O., Sur la mesure des ensembles plans dont tous les points sont rectilinêairement accessibles, Fund. Math., 10 (1927), 116–168.Google Scholar
5. Randolph, J.F., Some density properties of point sets, Ann. Math., 37 (1936), 336–344.Google Scholar