Geodesic
Controlled convergence theorems for Henstock-Kurzweil-Pettis integral on $m$-dimensional compact intervals
Czechoslovak Mathematical Journal, Tome 62 (2012) no. 1, pp. 243-255
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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.
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.
DOI : 10.1007/s10587-012-0009-6
Classification : 26A39, 28B05, 46G10
Keywords: Henstock-Kurzweil-Pettis integral; controlled convergence theorem; complete locally convex spaces; $m$-dimensional compact interval 
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Kaliaj, Sokol B.; Tato, Agron D.; Gumeni, Fatmir D. Controlled convergence theorems for Henstock-Kurzweil-Pettis integral on $m$-dimensional compact intervals. Czechoslovak Mathematical Journal, Tome 62 (2012) no. 1, pp. 243-255. doi: 10.1007/s10587-012-0009-6

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