Real Analysis Exchange

The dual of the Henstock-Kurzweil space

Genqian Liu

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We prove that if \(T\) is a continuous linear functional on the space \({\mathcal D}\) of Henstock-Kurzweil integrable functions on \([a_1,b_1] \times \cdots \times [a_m,b_m]\), then there exists a function \(g\) of strong bounded variation on \([a_1,b_1] \times \cdots \times [a_m,b_m]\) such that \[ T(f) = (HK) {\int \ldots \int \atop \scriptstyle{[a_1,b_1] \times \cdots \times [a_m,b_m]}} f(x_1,\ldots,x_m) g(x_1,\ldots,x_m)\, dx_1 \ldots dx_m\, . \]

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Real Anal. Exchange, Volume 22, Number 1 (1996), 105-121.

First available in Project Euclid: 1 June 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 26A39: Denjoy and Perron integrals, other special integrals 26A42: Integrals of Riemann, Stieltjes and Lebesgue type [See also 28-XX] 26B30: Absolutely continuous functions, functions of bounded variation

Linear functionals Henstock-Kurzweil integral strong bounded variation


Liu, Genqian. The dual of the Henstock-Kurzweil space. Real Anal. Exchange 22 (1996), no. 1, 105--121.

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