Learning decision rules for pattern classification under a family of probability measures

In this paper, the PAC learnability of decision rules for pattern classification under a family of probability measures is investigated. It is shown that uniform boundedness of the metric entropy of the class of decision rules is both necessary and sufficient for learnability if the family of probability measures is either compact, or contains an interior point, with respect to total variation metric. Then it is shown that learnability is preserved under finite unions of families of probability measures, and also that learnability with respect to each of a finite number of measures implies learnability with respect to the convex hull of the families of "commensurate" probability measures.<>