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On the relationship between differentiability and absolute continuity of measures on ℝn
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  • Published: June 1986

On the relationship between differentiability and absolute continuity of measures on ℝn

  • Denis Bell1 

Probability Theory and Related Fields volume 72, pages 417–424 (1986)Cite this article

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  • 4 Citations

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Summary

We give an elementary proof of the fact that a finite Borel measure on ℝn is absolutely continuous with a C 1 density if and only if it has directional derivatives which are continuous almost everywhere. The Radon-Nikodym derivative of a differentiable measure is given in terms of the directional derivatives.

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References

  1. Bell, D.: A quasi-invariance theorem for measures on Banach spaces. Trans. Am. Math. Soc. 290, 851–855 (1985)

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  2. Malliavin, P.: Stochastic calculus of variations and hypoelliptic operators. Proc. Intern. Sympos. S.D.E., Kyoto 1976

  3. Skorohod, A. V.: Integration in Hilbert space. Berlin Heidelberg New York: Springer 1974

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Authors and Affiliations

  1. Department of Mathematics, Suffolk University, 02114, Boston, MA, USA

    Denis Bell

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  1. Denis Bell
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Cite this article

Bell, D. On the relationship between differentiability and absolute continuity of measures on ℝn . Probab. Th. Rel. Fields 72, 417–424 (1986). https://doi.org/10.1007/BF00334194

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  • Received: 15 May 1985

  • Issue Date: June 1986

  • DOI: https://doi.org/10.1007/BF00334194

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Keywords

  • Stochastic Process
  • Probability Theory
  • Statistical Theory
  • Borel Measure
  • Directional Derivative
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