Can one decide the type of the mean from the empirical measure?

The problem of deciding whether the mean of an unknown distribution is in a set A or in its complement based on a sequence of independent random variables drawn according to this distribution is considered. We propose an algorithm which leads to an a.s. correct decision for any A in a class of sets satisfying certain structural assumptions. This class includes not only all countable sets, but many uncountable sets as well. A refined decision procedure is also presented which, given a countable decomposition of A, can determine a.s. to which set of the decomposition the mean belongs. This extends and simplifies a construction by Cover.