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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Integration theory on infinite-dimensional manifolds

Author: Hui Hsiung Kuo
Journal: Trans. Amer. Math. Soc. 159 (1971), 57-78
MSC: Primary 58B15; Secondary 28A40
MathSciNet review: 0295393
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Abstract: The purpose of this paper is to develop a natural integration theory over a suitable kind of infinite-dimensional manifold. The type of manifold we study is a curved analogue of an abstract Wiener space.

Let $ H$ be a real separable Hilbert space, $ B$ the completion of $ H$ with respect to a measurable norm and $ i$ the inclusion map from $ H$ into $ B$. The triple $ (i,H,B)$ is an abstract Wiener space. $ B$ carries a family of Wiener measures.

We will define a Riemann-Wiener manifold to be a triple $ (\mathcal{W},\tau ,g)$ satisfying specific conditions, $ \mathcal{W}$ is a $ {C^j}$-differentiable manifold $ (j \geqq 3)$ modelled on $ B$ and, for each $ x$ in $ \mathcal{W},\tau (x)$ is a norm on the tangent space $ {T_x}(\mathcal{W})$ of $ \mathcal{W}$ at $ x$ while $ g(x)$ is a densely defined inner product on $ {T_x}(\mathcal{W})$.

We show that each tangent space is an abstract Wiener space and there exists a spray on $ \mathcal{W}$ associated with $ g$. For each point $ x$ in $ \mathcal{W}$ the exponential map, defined by this spray, is a $ {C^{j - 2}}$-homeomorphism from a $ \tau (x)$-neighborhood of the origin in $ {T_x}(\mathcal{W})$ onto a neighborhood of $ x$ in $ \mathcal{W}$. We thereby induce from Wiener measures of $ {T_x}(\mathcal{W})$ a family of Borel measures $ {q_t}(x, \cdot ),t > 0$, in a neighborhood of $ x$. We prove that $ {q_t}(x, \cdot )$ and $ {q_s}(y, \cdot )$, as measures in their common domain, are equivalent if and only if $ t = s$ and $ {d_g}(x,y)$ is finite. Otherwise they are mutually singular. Here $ {d_g}$ is the almost-metric (in the sense that two points may have infinite distance) on $ \mathcal{W}$ determined by $ g$. In order to do this we first prove an infinite-dimensional analogue of the Jacobi theorem on transformation of Wiener integrals.

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Keywords: Integration theory, infinite-dimensional manifold, abstract Wiener space, Wiener measures, Jacobi Theorem, Riemann-Wiener manifold, Riemannian manifold, admissible transformation, spray, exponential map, Radon-Nikodym derivative, equivalence-perpendicularity dichotomy
Article copyright: © Copyright 1971 American Mathematical Society