The Stieltjes integrals in the theory of harmonic functions

We study various Stieltjes integrals, such as Poisson–Stieltjes, conjugate Poisson–Stieltjes, Schwartz–Stieltjes and Cauchy–Stieltjes, and prove theorems on the existence of their finite angular limits a.e. in terms of the Hilbert–Stieltjes integral. These results are valid for arbitrary bounded integrands that are differentiable a.e. and, in particular, for integrands of the class $\mathcal{CBV}$ (countably bounded variation).