On Denjoy-Lusin's theorem (Q2765936)

According to Men'shov (1916), a set \(E\subset[-\pi,\pi)\) is called an \(M\)-set in the narrow sense if there exists a nonnegative measure \(\mu\) such that \(\mu(S)= 0\) for all measurable sets \(S\) for which \(S\cap E= \emptyset\), \(\mu(E)= 1\), and NEWLINE\[NEWLINE\lim_{n\to \infty} \int^\pi_{-\pi} \cos nx d\mu(x)= \lim_{n\to\infty} \int^\pi_{-\pi} \sin nx d\mu(x)= 0.NEWLINE\]NEWLINE The author proves that if \(E\) is an \(M\)-set in the narrow sense and if NEWLINE\[NEWLINE\sum^\infty_{n=0} \rho_n|\cos(nx+ \theta_n)|^p< \infty\quad\text{for }x\in E,NEWLINE\]NEWLINE where \(\rho_n\geq 0\), \(p\) is a natural number and \(0\leq \theta_n< 2\pi\), then we necessarily have \(\sum \rho_n< \infty\).