The theorem is this: If $\mu$ is a positive regular Borel measure on the unit circle, then a closed subspace $\mathcal{M}$ of $L^2(\mu)$ satisfies $\chi_1 \mathcal{M}=\mathcal{M}$ if and only if there exist a Borel subset $E$ of the unit circle such that $\mathcal{M}=L_E^2(\mu)=\{f\in L^2(\mu):f(e^{it})=0 \text{ for } e^{it}\notin E\}$ where $\chi_1(e^{it})=e^{it}$.
In the proof, they show that the projection onto $\mathcal{M}$ is of the form $M_\phi$ for some $\phi\in L^\infty(\mu)$. Then they say that the result then follows. I am not seeing why the result then follows.