For a positive finite Borel measure $μ$ compactly supported in the complex plane, the space $\mathcal{P}^2(μ)$ is the closure of the analytic polynomials in the Lebesgue space $L^2(μ)$. According to Thomson's famous result, any space $\mathcal{P}^2(μ)$ decomposes as an orthogonal sum of pieces which are essentially analytic, and a residual $L^2$-space. We study the structure of this decomposition for a class of Borel measures $μ$ supported on the closed unit disk for which the part $μ_\mathbb{D}$, living in the open disk $\mathbb{D}$, is radial and decreases at least exponentially fast near the boundary of the disk. For the considered class of measures, we give a precise form of the Thomson decompsition. In particular, we confirm a conjecture of Kriete and MacCluer from 1990, which gives an analog to Szegö's classical theorem.