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.
References
Bell, D.: A quasi-invariance theorem for measures on Banach spaces. Trans. Am. Math. Soc. 290, 851–855 (1985)
Malliavin, P.: Stochastic calculus of variations and hypoelliptic operators. Proc. Intern. Sympos. S.D.E., Kyoto 1976
Skorohod, A. V.: Integration in Hilbert space. Berlin Heidelberg New York: Springer 1974
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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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DOI: https://doi.org/10.1007/BF00334194
Keywords
- Stochastic Process
- Probability Theory
- Statistical Theory
- Borel Measure
- Directional Derivative