  We present an exact derivation of a process in which a microscopic measured
system interacts with heat-bath and pointer modes of a measuring device, via a
coupling involving a general Hermitian operator $\Lambda$ of the system. In the
limit of strong interaction with these modes, over a small time interval, we
derive the exact effective many-body density matrix of the measured system plus
pointer. We then discuss the interpretation of the dynamics considered as the
first stage in the process of quantum measurement, eventually involving the
wave-function collapse due to interactions with "the rest of the universe." We
establish that the effective density matrix represents the required framework
for the measured system and the pointer part of the measuring device to evolve
into a statistical mixture described by direct-product states such that the
system is in each eigenstate of $\Lambda$ with the correct quantum-mechanical
probability, whereas the expectation values of pointer-space operators retain
amplified information of the system's eigenstate.
