  If ergodicity holds what can be shown? It can be shown that for all but a set of measure zero of initial points, the time average of a phase quantity over infinite time will equal its phase average. It can be shown that for any measurable region the average time the system spends in that region will be proportional to the region's size (as measured by the probability measure used in the microcanonical ensemble). A solution to a further problem is also advanced. Boltzmann knew that the standard probability distribution was invariant under time evolution given the dynamics of the systems. But how could we know that it was the only such invariant measure? With ergodicity we can show that the standard probability distribution is the only one that is so invariant, at least if we confine ourselves to probability measures that assign probability zero to every set assigned zero by the standard measure.