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On the construction of measures

Published online by Cambridge University Press:  26 February 2010

J. D. Knowles
Affiliation:
Westfield College, London, N.W.3.
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Extract

1. Given a metric space (X, ρ) a family of subsets of X which includes the empty set Ø, and a non-negative function τ on with τ(Ø)=0, an outer measure μ* may be defined by

where empty infimums have value +∞. It is easily seen that μ* is a metric outer measure [i.e., if ρ(A, B)>0 then μ*(AB)=μ*(A)+μ*(B)] and from this it follows that all Borel sets in X are μ*-measurable.

Type
Research Article
Copyright
Copyright © University College London 1966

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References

1. Munroe, M., Measure and integration (Addison-Wesley, 1953).Google Scholar
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4. Bledsoe, W. and Morse, A. P., “A topological measure construction”, Pacific J. of Math. 13 (1963), 10671076.Google Scholar