  We study the problem of lifting a measure to an induced map
$F(x)=f^{R(x)}(x)$. In particular, we give a necessary and sufficient condition
for an ergodic $f$ invariant probability $\mu$ to be $F$-liftable as well as a
condition for the lift to be an ergodic measure. Moreover, we show that every
lift of $\mu$ is a weighted average of the restriction of $\mu$ to a countable
number of $F$-ergodic components. We introduce the concept of a coherent
schedule of events and relate it to the lift problem. As a consequence, we
prove that we can always synchronize coherent schedules at almost every point
with respect to a given invariant probability $\mu$, showing that we can
synchronize `Pliss times' $\mu$ almost everywhere. We also provide a version of
this synchronization to non-invariant measures and, from that, we obtain some
results related to Viana's conjecture on the existence of SRB measures for maps
with non-zero Lyapunov exponents for Lebesgue almost every point.
