Boundary values in spaces spanned by rational functions and the index of invariant subspaces

Let (Formula presented.) be a positive compactly supported measure in the complex plane (Formula presented.), and for each (Formula presented.), let (Formula presented.) be the closed subspace of (Formula presented.) spanned by the polynomials. In 1991, Thomson gave a complete description of its structure, expressing (Formula presented.) as the direct sum of invariant subspaces, all but one of which is irreducible in the sense that it contains no non-trivial characteristic function. Years later, Aleman, Richter and Sundberg gave a more detailed analysis of the invariant subspaces in any irreducible summand. Here we discuss the extent to which those earlier results can be extended to (Formula presented.), the closed subspace of (Formula presented.) spanned by the rational functions having no poles on the support of (Formula presented.), by first establishing the existence of boundary values in these spaces. Our results all depend on the semiadditivity of analytic capacity, and ultimately on some form of the F. and M. Riesz theorem.