  Measurement in classical physics is examined here as a process involving the
joint evolution of object-system and measuring apparatus. For this, a model of
measurement is proposed which lends itself to theoretical analysis using
Hamiltonian mechanics and Bayesian probability. At odds with a widely-held
intuition, it is found that the ideal measurement capable of extracting finite
information without disturbing the system is ruled out (by the third law of
thermodynamics). And in its place a Heisenberg-like precision-disturbance
relation is found, with the role of $\hbar/2$ played by $k_BT/\Omega$; where
$T$ and $\Omega$ are a certain temperature and frequency characterizing the
ready-state of the apparatus. The proposed model is argued to be maximally
efficient, in that it saturates this Heisenberg-like inequality, while various
modifications of the model fail to saturate it. The process of continuous
measurement is then examined; yielding a novel pair of Liouville-like master
equations -- according to whether the measurement record is read or discarded
-- describing the dynamics of (a rational agent's knowledge of) a system under
continuous measurement. The master equation corresponding to discarded record
doubles as a description of an open thermodynamic system. The fine-grained
Shannon entropy is found to be a Lyapunov function (i.e. $\dot S\geq0$) of the
dynamics when the record is discarded, providing a novel H-theorem suitable for
studying the second law and non-equilibrium statistical physics. These findings
may also be of interest to those working on the foundations of quantum
mechanics, in particular along the lines of attempting to identify and unmix a
possible epistemic component of quantum theory from its ontic content. More
practically, these results may find applications in the fields of precision
measurement, nanoengineering and molecular machines.
