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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:
1
\(\{F_n\}_{n=1}^\infty\) is equiregulated.
2
\(\{F_n\}_{n=1}^\infty\) is uniformly \(g\)-normal.
3
\(\{F_n\}_{n=1}^\infty\) 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.

MSC:

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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