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