On Kurzweil-Stieltjes equiintegrability and generalized BV functions. (English) Zbl 07217263

The main result of the present paper provides sufficient conditions for Kurzweil-Stieltjes equiiintegrability of a sequence of functions \(\{f_n\}_{n=1}^\infty\) with respect to a function \(g:[a,b]\to\mathbb{R}\), which is regulated, left-continuous, and has generalized bounded variation.
It is assumed that the sequence \(\{f_n\}_{n=1}^\infty\) is pointwise bounded, the Kurzweil-Stieltjes integrals \(\int_a^b f_n\,\mathrm{d}g\) exist for all \(n\in\mathbb{N}\), and the indefinite integrals \(F_n(t)=\int_a^t f_n\,\mathrm{d}g\) have the following properties:
\(\{F_n\}_{n=1}^\infty\) is equiregulated.
\(\{F_n\}_{n=1}^\infty\) is uniformly \(g\)-normal.
\(\{F_n\}_{n=1}^\infty\) is uniformly \(g\)-differentiable (in the sense of Stieltjes derivatives) on \([a,b]\setminus Z\), where the variational measure of \(Z\) is zero.

It is shown that these conditions imply uniform integrability of \(\{f_n\}_{n=1}^\infty\) with respect to \(g\). Moreover, if \(\{f_n\}_{n=1}^\infty\) has a pointwise limit \(f\), then the integral \(\int_a^b f\,\mathrm{d}g\) exists and equals \(\lim_{n\to\infty}\int_a^b f_n\,\mathrm{d}g\).
The author also shows that the first and second condition of the main result cannot be dropped.


26A39 Denjoy and Perron integrals, other special integrals
26A42 Integrals of Riemann, Stieltjes and Lebesgue type
26A45 Functions of bounded variation, generalizations
26A24 Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems
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