I will discuss the following result: For a large class of function spaces, X, on the disk the boundary sets of X−capacity zero are exactly those which carry no non-trivial XCarleson measure. This gives a new approach to results such as Beurling’s classical result that functions in the Dirichlet space have radial limits on all radii outside an exceptional set of directions of capacity zero. I will also discuss how our approach, which is primarily geometric, adapts to classes of non-holomorphic functions such as harmonic or p−harmonic functions in n-space. (joint work with Nicola Arcozzi and Eric Sawyer)