  For strictly ergodic systems, we introduce the class of CF-Nil($k$) systems:
systems for which the maximal measurable and maximal topological $k$-step
pronilfactors coincide as measure-preserving systems. Weiss' theorem implies
that such systems are abundant in a precise sense. We show that the CF-Nil($k$)
systems are precisely the class of minimal systems for which the $k$-step
nilsequence version of the Wiener-Wintner average converges everywhere. As part
of the proof we establish that pronilsystems are $coalescent$ both in the
measurable and topological categories. In addition, we characterize a
CF-Nil($k$) system in terms of its $(k+1)$-$th\ dynamical\ cubespace$. In
particular, for $k=1$, this provides for strictly ergodic systems a new
condition equivalent to the property that every measurable eigenfunction has a
continuous version.
