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 $\left\{ x_{n}\right\} _{n\in I}$, and a measure over $\left( \mathbf{R}^{I},\mathcal{B}^{(I)}\right) $ 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)}\right) $. 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\right) $-general, with properties that generalize the analogous ones of the finite-dimensional diffeomorphisms