Measures on Compact HS Spaces

Overview & Synopsis

We construct two examples of a compact, 0-dimensional space which supports a Radon probability measure whose measure algebra is isomorphic to the measure algebra of $2^{\omega_1}$ . The first construction uses $\diamondsuit$ to produce an S-space with no convergent sequences in which every perfect set is a $G_\delta$. 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.