We define a generalized possibility measure (GPM) on a set X to be a function from the power set of X to the unit interval I = [0,1], whereby the measure of a finite union of subsets of X equals the maximum of their measures. We characterize GPMs as the infima of downward directed families of possibility measures. This helps to confer an intuitive meaning on GPMs. We discuss all directed families of possibility measures (or, equivalently, of possibility distributions) which generate a given GPM, and we identify the largest such family. We introduce extensions of GPMs to real functions on I X , obtained through a continuous triangular norm. We provide characterizations and alternative descriptions for those extensions.