  We study the classical problem of identifying the structure of $P^2(\mu)$,
the closure of analytic polynomials in the Lebesgue space $L^2(\mu)$ of a
compactly supported Borel measure $\mu$ living in the complex plane. In his
influential work, Thomson showed that the space decomposes into a full
$L^2$-space and other pieces which are essentially spaces of analytic functions
on domains in the plane. For a family of measures $\mu$ supported on the closed
unit disk $\overline{\mathbb{D}}$ which have a part on the open disk
$\mathbb{D}$ which is similar to the Lebesgue area measure, and a part on the
unit circle $\mathbb{T}$ which is the restriction of the Lebesgue linear
measure to a general measurable subset $E$ of $\mathbb{T}$, we extend the ideas
of Khrushchev and calculate the exact form of the Thomson decomposition of the
space $P^2(\mu)$. It turns out that the space splits according to a certain
decomposition of measurable subsets of $\mathbb{T}$ which we introduce. We
highlight applications to the theory of the Cauchy integral operator and de
Branges-Rovnyak spaces.
