  We propose a method, based on matrix product states, for studying the time
evolution of many-body quantum lattice systems under continuous and
site-resolved measurement. Both the frequency and the strength of generalized
measurements can be varied within our scheme, thus allowing us to explore the
corresponding two-dimensional phase diagram. The method is applied to
one-dimensional chains of nearest-neighbor interacting hard-core bosons. A
transition from an entangling to a disentangling (area-law) phase is found.
However, by resolving time-dependent density correlations in the monitored
system, we find important differences between different regions at the phase
boundary. In particular, we observe a peculiar phenomenon of
measurement-induced particle clusterization that takes place only for frequent
moderately strong measurements, but not for strong infrequent measurements.
