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 isomorphic to the measure algebra of ${\displaystyle 2^{{\omega }_1}}$. The first construction uses ♢ to produce an S-space with no convergent sequences in which every perfect set is a ${\displaystyle G_{\delta }}$. 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.