  We study countable partitions for measurable maps on measure spaces such that
for all point $x$ the set of points with the same itinerary of $x$ is
negligible. We prove that in nonatomic probability spaces every strong
generator (Parry, W., {\em Aperiodic transformations and generators}, J. London
Math. Soc. 43 (1968), 191--194) satisfies this property but not conversely. In
addition, measurable maps carrying partitions with this property are aperiodic
and their corresponding spaces are nonatomic. From this we obtain a
characterization of nonsingular countable to one mappings with these partitions
on nonatomic Lebesgue probability spaces as those having strong generators.
Furthermore, maps carrying these partitions include the ergodic
measure-preserving ones with positive entropy on probability spaces (thus
extending a result in Cadre, B., Jacob, P., {\em On pairwise sensitivity}, J.
Math. Anal. Appl. 309 (2005), no. 1, 375--382). Some applications are given.
