## On the double Lusin condition and convergence theorem for Kurzweil-Henstock type integrals.(English)Zbl 1389.26015

Let $$E\subset\mathbb R^n$$ be a closed nondegenerate interval and let $$\{f_{n}\}$$ be a sequence of Kurzweil-Henstock integrable real-valued functions on $$E$$. For each $$n$$, let $$F_n$$ denote an additive interval function defined by $F_n(I)=\int_If_n$ for each closed nondegenerate interval $$I$$ in $$E$$. Under this notation, recall that
(i)
$$\{f_n\}$$ is said to be equiintegrable on $$E$$ if for each $$\varepsilon>0$$ there is a gauge $$\delta$$ on $$E$$ such that, for all $$n$$, $(D)\sum| f_n (x)| I| -F_{n}(I)| <\varepsilon$ whenever $$D$$ is a $$\delta$$-fine partial division of $$E$$;
(ii)
$$\{f_{n}\}$$ is said to satisfy the uniform double Lusin condition (or $$\mathrm{UI}_1$$, in short) on $$E$$ if for each $$\varepsilon>0$$ there is a gauge $$\delta$$ on $$E$$ such that, for all $$n$$, $(D)\sum| f_n(x)| | I| <\varepsilon\text{ and }(D)\sum| F_n(I)| <\varepsilon$ whenever $$D$$ is a $$\delta$$-fine partial division of $$E$$ in $$\Gamma_{\varepsilon,n}$$, where $\Gamma_{\varepsilon,n}=\{(x,I):I\subset E, x\text{ is a vertex of }E\text{ and }| F_n(I)-f_n(x)| I| | \geqslant\varepsilon| I| \};$
(iii)
$$\{f_n\}$$ is said to satisfy the $$\mathrm{UI}_2$$ condition if for each $$\varepsilon>0$$ there is a gauge $$\delta$$ on $$E$$ such that, for all $$n$$, $(D)\sum| I| <\varepsilon\text{ and }(D)\sum| F_n(I)| <\varepsilon$ whenever $$D$$ is a $$\delta$$-fine partial division of $$E$$ in $$\Gamma_{\varepsilon,n}$$.
The key result of the paper under review, Theorem 3.3, compares the above three notions:
Suppose that $$f_n\rightarrow f$$ pointwise on $$E$$. Then, each of the above three statements about $$\{f_n\}$$ implies all the others, Kurzweil-Henstock integrability of the function $$f$$ on $$E$$, and the relation $\int_Ef=\lim_{n\rightarrow\infty}\int_Ef_n.$ In the remainder of the paper, the authors extend this result to the context of general division spaces.

### MSC:

 26A39 Denjoy and Perron integrals, other special integrals 28A20 Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence
Full Text: