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

Racca, Abraham; Cabral, Emmanuel

Math. Bohem. 141, No. 2, 153-168 (2016).

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.

Reviewer: Kirill Naralenkov (Moskva)

MSC:

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

Keywords:

Kurzweil-Henstock integral; equiintegrability; uniform double Lusin condition; division space