Geodesic
Conditions of convergence of boundary values of Cauchy type integrals
Matematičeskie zametki, Tome 5 (1969) no. 4, pp. 441-448

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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 $f_n(z)$. 
@article{MZM_1969_5_4_a6,
     author = {G. Ts. Tumarkin},
     title = {Conditions of convergence of boundary values of {Cauchy} type integrals},
     journal = {Matemati\v{c}eskie zametki},
     pages = {441--448},
     publisher = {mathdoc},
     volume = {5},
     number = {4},
     year = {1969},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1969_5_4_a6/}
}
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G. Ts. Tumarkin. Conditions of convergence of boundary values of Cauchy type integrals. Matematičeskie zametki, Tome 5 (1969) no. 4, pp. 441-448. http://geodesic.mathdoc.fr/item/MZM_1969_5_4_a6/