On existence conditions for the Stieltjes integral

A modification of the definition of the Stieltjes integral $\int_0^1f\,dg$ is proposed, and it is shown that this integral exists if $g\in\operatorname{Lip}\alpha$, $f\in W_1^{1-\alpha}$, and $0<\alpha<1$ ($W_1^{1-\alpha}$ is the Sobolev–Slobodetskii class. It is shown that this integral defines a general form of a linear functional on $W_1^{1-\alpha}$ and on the class $\operatorname{Lip}_0\alpha$ of functions $g$ for which $g(x)-g(y)=o(|x-y|^\alpha)$. Applications to the integration of abstract functions and to the theory of double operator integrals are given.
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