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A Condition of Halo Type for the Differentiation of Classes of Integrals

Published online by Cambridge University Press:  20 November 2018

C. A. Hayes Jr.
C. A. Hayes Jr.*
Affiliation:
University of California, Davis, California
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We shall consider a space S, a σ-algebra M of subsets of S, a measure μ defined on M, and the μ-integrals of certain μ-integrable functions f. To each point x of a certain set E of S we associate certain ones of the sets V ∈ M and form the quotients ∫ vf(x)dμ(x)/μ(V) for each such set V. In case these quotients tend to f(x) as the sets V converge to x in accordance with a definition we adopt in §2, then we say that the integral of f is differentiable or derivable at x. It is of interest to assert conditions that ensure the differentiability of a given integral or class of integrals at μ-almost all points of E.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 18 , 1966 , pp. 1015 - 1023
Copyright
Copyright © Canadian Mathematical Society 1966

References

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