  Let $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t \geq 0},\mathbb{P})$ be a
filtered probability space satisfying the usual assumptions: it is usually not
possible to extend to $\mathcal{F}_{\infty}$ (the $\sigma$-algebra generated by
$(\mathcal{F}_t)_{t \geq 0}$) a coherent family of probability measures
$(\mathbb{Q}_t)$, each of them being defined on $\mathcal{F}_t$. It is known
that for instance, on the Wiener space, this extension problem has a positive
answer if one takes the filtration generated by the coordinate process, but can
have a negative answer if one takes its usual augmentation. On the other hand,
the usual assumptions are crucial in order to obtain the existence of regular
versions of paths for most stochastic processes of interest, such as the local
time of the standard Brownian motion, stochastic integrals, etc. In order to
fix this problem, we introduce a new property for filtrations, intermediate
between the right continuity and the usual conditions. We show that most of the
important results of the theory of stochastic processes which are generally
proved under the usual augmentation, such as the existence of regular version
of trajectories or the d\'ebut theorem, still hold under the N-augmentation;
moreover this new augmentation allows the extension of a coherent family of
probability measures whenever this is possible with the original filtration.
