  We demonstrate that an attempt to measure a non-local in time quantity, such
as the time average $\la A\ra_T$ of a dynamical variable $A$, by separating
Feynman paths into ever narrower exclusive classes traps the system in
eigensubspaces of the corresponding operator $\a$. Conversely, in a long
measurement of $\la A\ra_T$ to a finite accuracy, the system explores its
Hilbert space and is driven to a universal steady state in which von Neumann
ensemble average of $\a$ coincides with $\la A\ra_T$. Both effects are
conveniently analysed in terms of singularities and critical points of the
corresponding amplitude distribution and the Zeno-like behaviour is shown to be
a consequence of conservation of probability.
