  As a full “rationale” for standard equilibrium statistical mechanics, however, much remains questionable. There is the problem that strict ergodicity is not true of realistic systems. There are many problems encountered if one tries to use the rationale as Boltzmann hoped to identify phase averages with measured quantities relying on the fact that macroscopic measurements take “long times” on a molecular scale. There are the problems introduced by the fact that all of the mathematically legitimate ergodic results are qualified by exceptions for “sets of measure zero.” What is it physically that makes it legitimate to ignore a set of trajectories just because it has measure zero in the standard measure? After all, such neglect leads to catastrophically wrong predictions when there really are hidden, global constants of motion. In proving the standard measure uniquely invariant, why are we entitled to ignore probability measures that assign non-zero probabilities to sets of conditions assigned probability zero in the standard measure? After all, it was just the use of that standard measure that we were trying to justify in the first place.