  In two recent papers we introduced some new techniques for constructing an
extension of a probability-preserving system $T:\mathbb{Z}^d\curvearrowright
(X,\mu)$ that enjoys certain desirable properties in connexion with the
asymptotic behaviour of some related nonconventional ergodic averages.
  The present paper is the first of two that will explore various refinements
and extensions of these ideas. This first part is dedicated to some much more
general machinery for the construction of extensions that can be used to
recover various earlier results. It also contains two relatively simple new
applications of this machinery to the study of certain families of
nonconventional averages, one in discrete and one in continuous time
(convergence being a new result for the latter).
  In the forthcoming second part (arXiv:0910.0907) we will introduce the
problem of describing the characteristic factors and the limit of the linear
nonconventional averages $\frac{1}{N}\sum_{n=1}^N \prod_{i=1}^kf_i\circ
T^{n\bf{p}_i}$ when the directions $\bf{p}_1$, $\bf{p}_2$, \ldots, $\bf{p}_k
\in \mathbb{Z}^d$ are not assumed to be linearly independent, and provide a
fairly detailed solution in the case when k = 3, d = 2 and any pair of
directions is linearly independent. This will then be used to prove the
convergence in $L^2(\mu)$ of the quadratic nonconventional averages
$\frac{1}{N}\sum_{n=1}^N (f_1\circ T_1^{n^2})(f_2\circ T_1^{n^2}T_2^n)$.
