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A sufficient condition for the differentiation of certain classes of set functions

Published online by Cambridge University Press:  24 October 2008

Charles A. Hayes Jr
Affiliation:
University of California Davis, California

Extract

We shall be concerned with the differentiation of an arbitrary set function ψ from certain classes of set functions with respect to another fixed set function φ. In (1), § 2·5, a halo evanescence condition is shown to be necessary and sufficient for the differentiation of a fixed measure ψ with respect to φ. Here we formulate a type of halo condition which ensures the differentiability of entire classes of set functions. Although the necessity of this condition is not shown, we demonstrate by means of an example that the results cannot be improved upon in certain respects.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1958

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References

REFERENCE

(1)Hayes, C. A. and Pauc, C. Y.Full individual and class differentiation theorems in their relation to halo and Vitali problems. Can. J. Math. 7 (1955), 221–74.CrossRefGoogle Scholar