Английская Википедия:Differentiable measure
Шаблон:Short description In functional analysis and measure theory, a differentiable measure is a measure that has a notion of a derivative. The theory of differentiable measure was introduced by Russian mathematician Sergei Fomin and proposed at the International Congress of Mathematicians in 1966 in Moscow as an infinite-dimensional analog of the theory of distributions.[1] Besides the notion of a derivative of a measure by Sergei Fomin there exists also one by Anatoliy Skorokhod,[2] one by Sergio Albeverio and Raphael Høegh-Krohn, and one by Oleg Smolyanov and Heinrich von Weizsäcker.[3]
Differentiable measure
Let
- <math>X</math> be a real vector space,
- <math>\mathcal{A}</math> be σ-algebra that is invariant under translation by vectors <math>h\in X</math>, i.e. <math>A +th\in \mathcal{A}</math> for all <math>A\in\mathcal{A}</math> and <math>t\in\R</math>.
This setting is rather general on purpose since for most definitions only linearity and measurability is needed. But usually one chooses <math>X</math> to be a real Hausdorff locally convex space with the Borel or cylindrical σ-algebra <math>\mathcal{A}</math>.
For a measure <math>\mu</math> let <math>\mu_h(A):=\mu(A+h)</math> denote the shifted measure by <math>h\in X</math>.
Fomin differentiability
A measure <math>\mu</math> on <math>(X,\mathcal{A})</math> is Fomin differentiable along <math>h\in X</math> if for every set <math>A\in\mathcal{A}</math> the limit
- <math>d_{h}\mu(A):=\lim\limits_{t\to 0}\frac{\mu(A+th)-\mu(A)}{t}</math>
exists. We call <math>d_{h}\mu</math> the Fomin derivative of <math>\mu</math>.
Equivalently, for all sets <math>A\in\mathcal{A}</math> is <math>f_{\mu}^{A,h}:t\mapsto \mu(A+th)</math> differentiable in <math>0</math>.[4]
Properties
- The Fomin derivative is again another measure and absolutely continuous with respect to <math>\mu</math>.
- Fomin differentiability can be directly extend to signed measures.
- Higher and mixed derivatives will be defined inductively <math>d^n_{h}=d_{h}(d^{n-1}_{h})</math>.
Skorokhod differentiability
Let <math>\mu</math> be a Baire measure and let <math>C_b(X)</math> be the space of bounded and continuous functions on <math>X</math>.
<math>\mu</math> is Skorokhod differentiable (or S-differentiable) along <math>h\in X</math> if a Baire measure <math>\nu</math> exists such that for all <math>f\in C_b(X)</math> the limit
- <math>\lim\limits_{t\to 0}\int_X\frac{ f(x-th)-f(x)}{t}\mu(dx)=\int_X f(x)\nu(dx)</math>
exists.
In shift notation
- <math>\lim\limits_{t\to 0}\int_X\frac{ f(x-th)-f(x)}{t}\mu(dx)=\lim\limits_{t\to 0}\int_Xf\; d\left(\frac{\mu_{th}-\mu}{t}\right).</math>
The measure <math>\nu</math> is called the Skorokhod derivative (or S-derivative or weak derivative) of <math>\mu</math> along <math>h\in X</math> and is unique.[4][5]
Albeverio-Høegh-Krohn Differentiability
A measure <math>\mu</math> is Albeverio-Høegh-Krohn differentiable (or AHK differentiable) along <math>h\in X</math> if a measure <math>\lambda\geq 0</math> exists such that
- <math>\mu_{th}</math> is absolutely continuous with respect to <math>\lambda</math> such that <math>\lambda_{th}=f_t\cdot \lambda</math>,
- the map <math>g:\R\to L^2(\lambda),\; t\mapsto f_{t}^{1/2}</math> is differentiable.[4]
Properties
- The AHK differentiability can also be extende to signed measures.
Example
Let <math>\mu</math> be a measure with a continuously differentiable Radon-Nikodým density <math>g</math>, then the Fomin derivative is
- <math>d_{h}\mu(A)=\lim\limits_{t\to 0}\frac{\mu(A+th)-\mu(A)}{t}=\lim\limits_{t\to 0}\int_A\frac{g(x+th)-g(x)}{t}\mathrm{d}x=\int_A g'(x)\mathrm{d}x.</math>
Bibliography
- Шаблон:Cite book
- Шаблон:Cite journal
- Шаблон:Cite book
- Шаблон:Cite conference
- Kuo, Hui-Hsiung “Differentiable Measures.” Chinese Journal of Mathematics 2, no. 2 (1974): 189–99. Шаблон:JSTOR.
References
