Differentiable measure

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. Besides the notion of a derivative of a measure by Sergei Fomin there exists also one by Anatoliy Skorokhod, one by Sergio Albeverio and Raphael Høegh-Krohn, and one by Oleg Smolyanov and Heinrich von Weizsäcker.

Differentiable measure
Let
 * $$X$$ be a real vector space,
 * $$\mathcal{A}$$ be σ-algebra that is invariant under translation by vectors $$h\in X$$, i.e. $$A +th\in \mathcal{A}$$ for all $$A\in\mathcal{A}$$ and $$t\in\R$$.

This setting is rather general on purpose since for most definitions only linearity and measurability is needed. But usually one chooses $$X$$ to be a real Hausdorff locally convex space with the Borel or cylindrical σ-algebra $$\mathcal{A}$$.

For a measure $$\mu$$ let $$\mu_h(A):=\mu(A+h)$$ denote the shifted measure by $$h\in X$$.

Fomin differentiability
A measure $$\mu$$ on $$(X,\mathcal{A})$$ is Fomin differentiable along $$h\in X$$ if for every set $$A\in\mathcal{A}$$ the limit
 * $$d_{h}\mu(A):=\lim\limits_{t\to 0}\frac{\mu(A+th)-\mu(A)}{t}$$

exists. We call $$d_{h}\mu$$ the Fomin derivative of $$\mu$$.

Equivalently, for all sets $$A\in\mathcal{A}$$ is $$f_{\mu}^{A,h}:t\mapsto \mu(A+th)$$ differentiable in $$0$$.

Properties

 * The Fomin derivative is again another measure and absolutely continuous with respect to $$\mu$$.
 * Fomin differentiability can be directly extend to signed measures.
 * Higher and mixed derivatives will be defined inductively $$d^n_{h}=d_{h}(d^{n-1}_{h})$$.

Skorokhod differentiability
Let $$\mu$$ be a Baire measure and let $$C_b(X)$$ be the space of bounded and continuous functions on $$X$$.

$$\mu$$ is Skorokhod differentiable (or S-differentiable) along $$h\in X$$ if a Baire measure $$\nu$$ exists such that for all $$f\in C_b(X)$$ the limit
 * $$\lim\limits_{t\to 0}\int_X\frac{ f(x-th)-f(x)}{t}\mu(dx)=\int_X f(x)\nu(dx)$$

exists.

In shift notation
 * $$\lim\limits_{t\to 0}\int_X\frac{ f(x-th)-f(x)}{t}\mu(dx)=\lim\limits_{t\to 0}\int_Xf\; d\left(\frac{\mu_{th}-\mu}{t}\right).$$

The measure $$\nu$$ is called the Skorokhod derivative (or S-derivative or weak derivative) of $$\mu$$ along $$h\in X$$ and is unique.

Albeverio-Høegh-Krohn Differentiability
A measure $$\mu$$ is Albeverio-Høegh-Krohn differentiable (or AHK differentiable) along $$h\in X$$ if a measure $$\lambda\geq 0$$ exists such that
 * 1) $$\mu_{th}$$ is absolutely continuous with respect to $$\lambda$$ such that $$\lambda_{th}=f_t\cdot \lambda$$,
 * 2) the map $$g:\R\to L^2(\lambda),\; t\mapsto f_{t}^{1/2}$$ is differentiable.

Properties

 * The AHK differentiability can also be extended to signed measures.

Example
Let $$\mu$$ be a measure with a continuously differentiable Radon-Nikodým density $$g$$, then the Fomin derivative is
 * $$d_{h}\mu(A)=\lim\limits_{t\to 0}\frac{\mu(A+th)-\mu(A)}{t}=\lim\limits_{t\to 0}\int_A\frac{g(x+th)-g(x)}{t}\mathrm{d}x=\int_A g'(x)\mathrm{d}x.$$