Integral representation of linear functionals on vector lattices and its application to BV functions on Wiener space.

*(English)*Zbl 1075.60059
Kunita, Hiroshi (ed.) et al., Stochastic analysis and related topics in Kyoto. In honour of Kiyoshi ItĂ´. Lectures given at the conference, Kyoto, Japan, September 4–7, 2002. Tokyo: Mathematical Society of Japan (ISBN 4-931469-26-4/hbk). Advanced Studies in Pure Mathematics 41, 121-140 (2004).

If \(X\) is a compact Hausdorff space, then the Riesz representation theorem says that any bounded linear functional \(T\) on \(C(X)\) can be expressed as \(T(v)= \int_X v(x) d\nu(x)\) for some finite signed measure \(\nu\). A similar result holds for \(T(v)={\mathcal E}(u,v)\) for a quasi-regular Dirichlet form \(({\mathcal E},{\mathcal F})\). The main purpose of this paper is to generalize this result to the case of vector lattice by an analytic method. Let \(X\) be a topological space and \(\lambda\) a Borel measure on \(X\). Consider a separable uniformly convex Banach space \((D,\| \cdot\| _D)\) constituting from a subspace of Borel measurable functions on \(X\). Further assume that any sequence converging to 0 in \(D\) contains a subsequence which converges to 0 \(\lambda\)-a.s. Similarly to the case of Dirichlet forms, assume that \(D\) satisfies the normal contraction and quasi-regularity properties. Choosing suitable functions \(\rho \in D\) and \(\xi\) on \([0,\infty)\), a capacity related to \(D\) is introduced. For this capacity, the notion of nest is defined as in the case of quasi-regular Dirichlet forms. Then the main theorem of this paper says that the following two statements are equivalent. (i) There exists a nest \(\{F_k\}\) and constants \(\{C_k\}\) such that \(T(v) \leq C_k \| v\| _{L^\infty(X)}, \;\forall v \in D_{b,F_k}\). (ii) There exists a finite signed smooth measure \(\nu\) and a nest \(\{F_f'\}\) such that \(T(v)=\int_X \tilde{v}(z)\nu(dz), \;\forall v \in D_{b,F_k'}\), where \(D_A\) is the family of elements of \(D\) which vanishes on \(X\setminus A\) a.e. This result is applied to a function of bounded variation on Wiener space to improve the proof of the smoothness of the associated measure. Furthermore a nontrivial example of function of bounded variation is also given.

For the entire collection see [Zbl 1050.60002].

For the entire collection see [Zbl 1050.60002].

Reviewer: Yoichi Oshima (Kumamoto)