We study the classical problem of identifying the structure of , the closure of analytic polynomials in the Lebesgue space of a compactly supported Borel measure living in the complex plane. In his influential work, Thomson [Ann. of Math. (2) 133 (1991), pp. 477–507] showed that the space decomposes into a full -space and other pieces which are essentially spaces of analytic functions on domains in the plane. For a family of measures supported on the closed unit diskwhich have a part on the open disk which is similar to the Lebesgue area measure, and a part on the unit circle which is the restriction of the Lebesgue linear measure to a general measurable subset of , we extend the ideas of Khrushchev and calculate the exact form of the Thomson decomposition of the space . It turns out that the space splits according to a certain natural decomposition of measurable subsets of which we introduce. We highlight applications to the theory of the Cauchy integral operator and de Branges-Rovnyak spaces.