Regularity points and Jensen measures for $R(X)$
We discuss two types of “regularity point”, points of continuity and R-points for Banach function algebras, which were introduced by the first author and Somerset (2000). We show that, even for the natural uniform algebras $R(X)$ (for compact plane sets $X$), these two types of regularity point can be different. We then give a new method for constructing Swiss cheese sets $X$ such that $R(X)$ is not regular, but has no non-trivial Jensen measures. The original construction appears in the first author’s previous work. Our new approach to constructing such sets is more general, and allows us to obtain additional properties. In particular, we use our construction to give an example of such a Swiss cheese set $X$ with the property that the set of points of discontinuity for $R(X)$ has positive area.