Change of variables' formula for the integration of the measurable real functions over infinite-dimensional Banach spaces

In this paper we study, for any subset $I$ of $mathbf{N}^{ast}$ and for any strictly positive integer $k$, the Banach space $E_{I}$ of the bounded real sequences $left{ x_{n} ight} _{nin I}$, and a measure over $left( mathbf{R}^{I},mathcal{B}^{(I)} ight) $ that generalizes the $k$-dimensional Lebesgue one. Moreover, we recall the main results about the differentiation theory over $E_{I}$. The main result of our paper is a change of variables' formula for the integration of the measurable real functions on $left( mathbf{R}^{I},mathcal{B}^{(I)} ight) $. This change of variables is defined by some functions over an open subset of $E_{J}$, with values on $E_{I}$, called $left( m,sigma ight) $-general, with properties that generalize the analogous ones of the finite-dimensional diffeomorphisms.