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.
