Integration theory on infinite-dimensional manifolds

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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 (/, 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 r, g) satisfying specific conditions. iK is a C'-differentiable manifold (y’^3) modelled on B and, for each x in iK, t(x) is a norm on the tangent space Tx{?T) of iK at x while #(*) is a densely defined inner product on Tx(iK). We show that each tangent space is an abstract Wiener space and there exists a spray on iK associated with g. For each point * in iP the exponential map, defined by this spray, is a C'˜2-homeomorphism from a r(;c)-neighborhood of the origin in Tx(iK) onto a neighborhood of x in iK. We thereby induce from Wiener measures of Tx(vV) a family of Borel measures qt(x, •)> />0, in a neighborhood of x. We prove that qt(x, 0) and qs(y, 0)> as measures in their common domain, are equivalent if and only if t = s and d9(x, y) is finite. Otherwise they are mutually singular. Here d9 is the almost-metric (in the sense that two points may have infinite distance) on iK determined by g. In order to do this we first prove an infinite-dimensional analogue of the Jacobi theorem on transformation of Wiener integrals. © 1971 American Mathematical Society.

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Transactions of the American Mathematical Society

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