  We explore the effects of continuous number density measurement on atoms in
an optical lattice. By integrating a master equation for quantum observables,
we calculate how single particle correlations decay. We consider weakly- and
strongly- interacting bosons and noninteracting fermions. Even in the Mott
regime, such measurements destroy correlations and increase the average energy,
as long as some hopping is allowed. We explore the role of spatial resolution,
and find that the heating rate is proportional to the amount of information
gained from such measurements.
