  We study a Bose-Einstein condensate (BEC) in a double-well potential subject
to an unsharp continuous measurement of the atom number in one of the two
wells. We investigate the back action of the measurement on the quantum
dynamics and the viability to monitor the ensuing time evolution. For vanishing
inter-atomic interactions, mainly the expectation values of the measured local
observable can be inferred from the measurement record. Conversely, in the
presence of moderate inter-atomic interactions, the entire many-body state
--modified by the measurement-- is monitored with unit fidelity and, at the
same time, the measurement effects a transition from quantum to mean-field
(classical) behavior of the BEC. We show that this perfect state estimation is
possible because the inter-atomic interactions enhance the information gained
via the measurement.
