The limit set of the Henstock-Kurzweil integral sums of a vector-valued function
Abstract
We introduce the notion of the limit set $I_{H-K}(f)$ of the Henstock-Kurzweil integral sums of a function $f: [0, 1]\to X$, where $X$ is a Banach space, and study its properties. In particular, we construct an example of function $f$, which is not integrable, but its limit set consists exactly of one point. We find sufficient conditions that guarantee the convexity of the limit set.
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References
H.W. Ellis, On the limits of Riemann sums, J. London Math. Soc. 34 (1959), - no. 1, - P. 93-100.
S. Heikkila, Di erential and integral equations with Henstock-Kurzweil integrable functions, J. Math. Anal. Appl. 379. - 2011. - P. 171-179.
M. Nakamura, I. Amemiya, On the limit of Riemann sums of functions in Banach spaces, J. Fac. Sci. Hokkaido Univ. Ser. I 19 (1966), - no. 3.4, - P. 135-145.
M.I. Kadets, V.M. Kadets, Series in Banach spaces. Conditional and unconditional convergence, Birkhauser, Basel - Boston - Berlin, 1997. - 156 p.
V.M. Kadets, The starness of the domain of limits of Riemann integral sums of a vector-valued function, in: Operator Theory, Subharmonic Functions, - Naukova Dumka, - Kiev, - 1991. - P. 60-67 (Russian).
V.M. Kadets, M.I. Kadets, Conditions for the convexity of the set of limits of Riemann sums of a vector-valued function, Mat. Zametki 35 (1984), - no. 2, - P. 161-167 (Russian).
J.L. Kelley, General Topology, - Springer, - 1975. - 298 p.
D.S. Kurtz, Ch.W. Swartz. Series in real analysis - volume 9: theories of integration. The integrals of Riemann, Lebesgue, Henstock-Kurzweil and McShane, - World scienti c, - 2004. - 268 p.
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