Integration over an Infinite-Dimensional Banach Space and Probabilistic Applications

We study, for some subsets of , the Banach space of bounded real sequences . For any integer , we introduce a measure over that generalizes the -dimensional Lebesgue measure; consequently, also a theory of integration is defined. The main result of our paper is a change of variables' formula for the integration. 1. Introduction In the mathematical literature, some articles introduced infinite-dimensional measures analogous to the Lebesgue one (see, e.g., the paper of Léandre [1], in the context of the noncommutative geometry, that one of Tsilevich et al. [2], which studies a family of -finite measures on , and that one of Baker [3], which defines a measure on that is not -finite). The motivation of this paper follows from the natural extension to the infinite-dimensional case of the results of the article [4], where we estimate the rate of convergence of some Markov chains in to a uniform random vector. In order to consider the analogue random elements in , it is necessary to overcome some difficulties, for example, the lack of a change of variables’ formula for the integration in the subsets of . A related problem is studied in the paper of Accardi et al. [5], where the authors describe the transformations of generalized measures on locally convex spaces under smooth transformations of these spaces. In our paper, we consider some subsets of , and we suppose that is endowed with the standard infinity-norm generalized to assume the values in ; then, the vector space of the elements of with finite norm is a Banach space with respect to the distance defined by the norm. Observe that although in general it is possible to construct a -algebra on simply by considering the product indexed by of the same Borel -algebra on , in this way a product of -finite measures on can be defined only if is finite or is a probability measure (by Jessen theorem). To solve this problem and others, in Section 2 we use Corollary 4 (that generalizes the Jessen theorem) to define a measure over , where ; consequently, we define also a theory of integration. In the case , the measure coincides with the -dimensional Lebesgue measure on . In Section 3, we introduce the determinant of a class of infinite-dimensional matrices, called -standard, and we expose briefly a theory that generalizes the standard theory of the matrices. Moreover, we prove that the determinant of a -standard matrix is equal to the product of its eigenvalues, as in the finite-dimensional case. In Section 4, we present the main result of our paper, that is, a change of variables formula for the integration of the