Inversion formulae for the probability measures on Banach spaces

Abstract:

Let $B$ be a real separable Banach space, and let $\mu$ be a probability measure on $\mathcal {B}(B)$, the Borel sets of $B$. The characteristic functional (Fourier transform) $\phi$ of $\mu$, defined by $\phi (y) = \int _B {\exp \{ i(y,x)\} d\mu (x)\;}$ for $y \in {B^\ast }$ (the topological dual of $B$), uniquely determines $\mu$. In order to determine $\mu$ on $\mathcal {B}(B)$, it suffices to obtain the value of $\int _B {G(s)d\mu (s)}$ for every real-valued bounded continuous function $G$ on $B$. Hence an inversion formula for $\mu$ in terms of $\phi$ is obtained if one can uniquely determine the value of $\int _B {G(s)d\mu (s)}$ for all real-valued bounded continuous functions $G$ on $B$ in terms of $\phi$ and $G$. The main efforts of this paper will be to prove such inversion formulae of various types. For the Orlicz space ${E_\alpha }$ of real sequences we establish inversion formulae (Main Theorem II) which properly generalize the work of L. Gross and derive as a corollary the extension of the Main Theorem of L. Gross to ${E_\alpha }$ spaces (Corollary 2.2.12). In Part I we prove a theorem (Main Theorem I) which is Banach space generalization of the Main Theorem of L. Gross by reinterpreting his necessary and sufficient conditions in terms of convergence of Gaussian measures. Finally, in Part III we assume our Banach space to have a shrinking Schauder basis to prove inversion formulae (Main Theorem III) which express the measure directly in terms of $\phi$ and $G$ without the use of extension of $\phi$ as required in the Main Theorems I and II. Furthermore this can be achieved without using the Lévy Continuity Theorem and we hope that one can use this theorem to obtain a direct proof of the Lévy Continuity Theorem.