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Keywords:
Henstock-Kurzweil-Pettis integral; controlled convergence theorem; complete locally convex spaces; $m$-dimensional compact interval
Summary:
In this paper we use a generalized version of absolute continuity defined by J. Kurzweil, J. Jarník, Equiintegrability and controlled convergence of Perron-type integrable functions, Real Anal. Exch. 17 (1992), 110–139. By applying uniformly this generalized version of absolute continuity to the primitives of the Henstock-Kurzweil-Pettis integrable functions, we obtain controlled convergence theorems for the Henstock-Kurzweil-Pettis integral. First, we present a controlled convergence theorem for Henstock-Kurzweil-Pettis integral of functions defined on $m$-dimensional compact intervals of $\mathbb {R}^{m}$ and taking values in a Banach space. Then, we extend this theorem to complete locally convex topological vector spaces.
References:
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