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 Alpha or in its complement based on a sequence of independent random variables drawn according to this distribution is considered. Using large deviations techniques, an algorithm is proposed which is shown to lead to an a.s. correct decision for a class of Alpha which are necessarily countable. A refined decision procedure is also presented which, given a countable decomposition of Alpha, can determine a.s. to which set of the decomposition the mean belongs. This extends and simplifies a construction by Cover.