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 .
Differentiable measure Let be a real vector space, be σ-algebra that is invariant under translation by vectors , i.e. for all and .
This setting is rather general on purpose since for most definitions only linearity and measurability is needed. But usually one chooses to be a real Hausdorff locally convex space with the Borel or cylindrical σ-algebra .
For a measure let denote the shifted measure by .
Fomin differentiability A measure on is Fomin differentiable along if for every set the limit
exists. We call the Fomin derivative of .
Equivalently, for all sets is differentiable in .
Properties The Fomin derivative is again another measure and absolutely continuous with respect to . Fomin differentiability can be directly extend to signed measures. Higher and mixed derivatives will be defined inductively .
Skorokhod differentiability Let be a Baire measure and let be the space of bounded and continuous functions on .
is Skorokhod differentiable (or S-differentiable) along if a Baire measure exists such that for all the limit
exists.
In shift notation
The measure is called the Skorokhod derivative (or S-derivative or weak derivative) of along and is unique.
Albeverio-Høegh-Krohn Differentiability A measure is Albeverio-Høegh-Krohn differentiable (or AHK differentiable) along if a measure exists such that is absolutely continuous with respect to such that , the map is differentiable.
Properties The AHK differentiability can also be extended to signed measures.
Example Let be a measure with a continuously differentiable Radon-Nikodým density , then the Fomin derivative is
Bibliography
Kuo, Hui-Hsiung “Differentiable Measures.” Chinese Journal of Mathematics 2, no. 2 (1974): 189–99. .
References