## 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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