Generating the k-Subsets of an n-Set

Abstract

While discussing with colleagues certain questions about generalized measures on finite measure spaces, the author came upon a combinatorial lemma-on generating all of the k-subsets of an n-set. In the context of these discussions, one is given an n-set, say X= {x,x2, ... .,xn,, together with a collection 9 of its subsets that is closed under disjoint union and under complementation. A generalized measure It is a nonnegative real-valued function on these subsets, one however that is additive over disjoint unions. Of course, there are any number of such generalized measure spaces (g.m.s.), some interesting and some not, depending on one's point of view. Two useful but quite different introductions to the subject are those of Rado [1] and Gudder [2]. In the present context, we would like to be assured, first of all, that there are generalized measures that are not induced by an ordinary signed measure on the points of X. For this purpose, we let X = (1,2,3,4,5,6), and as the measurable sets we take