A Stone Space Approach to the Existence of Bounded Common Extensions

Let A and B be fields of subsets of a set X and let m: AªR and n : BªR be bounded, finitely additive measures i.e., charges.. We give global necessary and sufficient conditions on A and B under which any bounded, consistent charges m and n have a bounded common extension. Conditions on A and B are given under which any 0]1 valued charges m and n have an integer valued common extension. Our results, together with Example 4.4 infra, serve to answer open questions left unresolved by A. Basile, K. P. S. Bhaskara Rao, and R. M. Shortt Proc. Amer. Math. Soc. 121, 1994, 137]143., and thereby end a line of enquiry begun by Z. Lipecki Czechoslo¨ak Math. J. 36, 1986, 489]494..