Measures on compact HS spaces

We construct two examples of a compact, 0-dimensional space which supports a Radon probability measure whose measure algebra is isomor-phic to the measure algebra of 2 ! 1. The rst construction uses } to produce an S-space with no convergent sequences in which every perfect set is a G. A space with these properties must be both hereditarily normal and hereditarily countably paracompact. The second space is constructed under CH and is both HS and HL. x0. Introduction. All spaces considered here are Hausdorr. A perfect set is a non-empty closed set with no isolated points. Suppose X is compact and supports a Radon probability measure such that the measure algebra of X; is not separable; does this imply that X can be mapped continuously onto 0; 1] ! 1 ? This question is open in ZFC. In particular, Haydon asked whether such an implication might follow from something like MA+:CH; see Fremlin 3] for more discussion. Under CH, there is a counterexample which is, in addition, a compact L-space (hereditarily Lindell of (HL) but not hereditarily separable (HS)); see 1,5,7]. In this paper, we show that, assuming }, there is another counterexample which is an S-space (HS, but not HL). The space also has the property that every perfect set is a G , whereas no point is a G. Also, assuming just CH, we construct a third counterexample which is both HS and HL. Neither of the above mentioned examples could be constructed in ZFC, since under MA + :CH, there are neither compact L-spaces (Juhh asz) nor compact S-spaces (Szent-mikll ossy) (see 9]). Furthermore, under MA + :CH, the measure algebra of any compact HL (equivalently, HS) Radon measure space is separable (Fremlin 3]). The following theorem details the properties of the S-space. The HS + HL example is a modiication of the S-space, and is described in x4.