  Measurements on classical systems are usually idealized and assumed to have
infinite precision. In practice, however, any measurement has a finite
resolution. We investigate the theory of non-ideal measurements in classical
mechanics using a measurement probe with finite resolution. We use the von
Neumann interaction model to represent the interaction between system and
probe. We find that in reality classical systems are affected by measurement in
a similar manner as quantum systems. In particular, we derive classical
equivalents of L\"uders' rule, the "collapse postulate", and the Lindblad
equation.
