On the existence of the Stieltjes integral for functions of bounded variation

Abstract

We obtain sufficient conditions of existence of the Stieltjes integral $$\int\limits_s^t {f(\tau )} d\mathcal{F}(\tau ) = \mathop {\lim }\limits_{\delta _n \to 0} \sum\limits_{k = 1}^{m_n } {f(\xi _k )(\mathcal{F}(t_k^n ) - \mathcal{F}(t_{k - 1}^n ))}$$ for functions of bounded variation taking values in a Banach algebra with identity regardless of the choice of points $ξ_k \in [t_{k−1}, t_k]$.