On the relationship between differentiability and absolute continuity of measures on \({\mathbb{R}}^ n\) (Q1065956)
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scientific article; zbMATH DE number 3923030
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | On the relationship between differentiability and absolute continuity of measures on \({\mathbb{R}}^ n\) |
scientific article; zbMATH DE number 3923030 |
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On the relationship between differentiability and absolute continuity of measures on \({\mathbb{R}}^ n\) (English)
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1986
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We give an elementary proof of the fact that a finite Borel measure on \({\mathbb{R}}^ n\) is absolutely continuous with a \(C^ 1\) density if and only if it has directional derivatives which are continuous almost everywhere. The Radon-Nikodým derivative of a differentiable measure is given in terms of the directional derivatives.
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finite Borel measure
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directional derivatives
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Radon-Nikodým derivative
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differentiable measure
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absolute continuity
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