Let $\mu$ be a complex measure on the unit circle. The Wiener theorem says that the sequence of the Cesaro means of $|\hat\mu_n|$ has a limit. Define $p_n(z)=\sum_{k=0}^n \hat\mu_k z^k$. Then the Abel means of $p_n$ have limits at almost all points of the unit circle. My question is: Are there any facts about the averaged convergence of $|p_n|^2$?