 Simply neglecting sets of measure zero seems to be problematic for various reasons. First, sets of measure zero can be rather ‘big’; for instance, the rational numbers have measure zero within the real numbers. Moreover, a set of measure zero need not be (or even appear) negligible if sets are compared with respect to properties other than their measures. For instance, we can judge the ‘size’ of a set by its cardinality or Baire category rather than by its measure, which leads us to different conclusions about the set’s size (Sklar 1993, pp. 182–88). It is also a mistake to assume that an event with measure zero cannot occur. In fact, having measure zero and being impossible are distinct notions. Whether or not the system at some point was in one of the special initial conditions for which the space and time mean fail to be equal is a factual question that cannot be settled by appeal to measures; pointing out that such points are scarce in the sense of measure theory does resolve the problem because it does not imply that they are scarce in the world as well.