Some Remarks on Real-valued Measurable Cardinals

We consider the set [u>]u and the cofinality of the set KX assuming that some cardinals are endowed in total measures. Introduction. A cardinal k is real-valued measurable if there exists a /ccomplete atomless probability measure on P(k), the set of all subsets of the cardinal /c. The status and the philosophy concerned e.g. real-valued measurable cardinals have been detailed and presented in a survey paper by A. Kanamori and M. Magidor [KM]. We shall concentrate mainly on two problems: the cofinality of sets of functions with respect to eventual domination and some combinatorics on u>, both assuming the existence of some total measures. T. Jech and K. Prikry [JP] showed, assuming 2W is real-valued measurable, that the cofinality of the set of all functions from u>i into to equals 2". We extend and complete their result. We show (Theorems 1 and 3, §2), under the same assumption about 2W, that if k < 2W is regular and w < A < k, then the cofinality of the set of all functions from k into A equals 2W provided cf(A) = w and the cofinality is less than 2W provided cf (A) > u. It is well known that under Martin's axiom, 2W cannot be real-valued measurable. The reason is, that under Martin's axiom the cofinality of the set of all functions from uj into w equals 2U while assuming 2W is real-valued measurable, the cofinality is < 2W (see also Theorem 4, §2). We give some other reasons for which some cardinals cannot carry total measures. We show (Corollary 2, §1) that if there exists a maximal /c-tower on u, then k is not real-valued measurable cardinal. It has been shown by S. Hechler [H] that each regular cardinal /c, w < k < 2U can be (consistently) the length of some maximal fc-tower on w. Throughout the paper we use standard set-theoretical notation. For example [u]u is used to denote the set of all infinite subsets of the least infinite ordinal w. All undefined terms can be found in [J]. 1. Let uw denote the set of all functions from ui into ui. For two arbitrary functions f,g G ww we set / <* g iff f(n) < g(n) for all but finitely many n G w. Received by the editors July 30, 1986 and, in revised form, November 20, 1987. Presented at Spring Topology Conference, Lafayette, April 3-5, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 03E55, 04A20.