Conditions of convergence of boundary values of Cauchy type integrals

In a domain $G$ bounded by a rectifiable Jordan curve $\gamma$ let be given a sequence of analytic functions $\{f_n(z)\}$ representable by Cauchy–Lebesgue type integrals
$$ f_n(z)=\int_\gamma\frac{\omega_n(\zeta)}{\zeta-z}d\zeta. $$
A theorem is established which enables one to determine from the convergence in measure of $\{\omega_n(\zeta)\}$ on a set $e\subset\gamma$ whether or not there is convergence in measure on the same set of $\{f_n(\zeta)\}$, the angular boundary values of the functions $f_n(z)$.