Hostname: page-component-77c89778f8-fv566 Total loading time: 0 Render date: 2024-07-23T03:21:29.890Z Has data issue: false hasContentIssue false
  • English
  • Français

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
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.

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

1. Burkill, J. C., On the differentiability of multiple integrals, J. London Math. Soc., 26 (1951), 244–249.Google Scholar
2. Busemann, H. and Feller, W., Zur Differentiation der Lebesgueschen Integrale, Fund. Math., 22 (1934), 226–256.Google Scholar
3. Halmos, P., Measure theory (New York, 1950).Google Scholar
4. Hayes, C. and Pauc, C. Y., Full individual and class differentiation theorems in their relations to halo and Vitali properties, Can. J. Math., 7 (1955), 221–274.Google Scholar
5. Jessen, B., Marcinkiewicz, J. and Zygmund, A., Note on the differentiability of multiple integrals, Fund. Math., 25 (1935), 217–234.Google Scholar