Product variational measures and Fubini–Tonelli type theorems for the Henstock–Kurzweil integral II

Abstract

This paper is a continuation of the paper [T.Y. Lee, Product variational measures and Fubini-Tonelli type theorems for the Henstock-Kurzweil integral, J. Math. Anal. Appl. 298 (2004) 677-692], in which we proved several Fubini-Tonelli type theorems for the Henstock-Kurzweil integral. Let f be Henstock-Kurzweil integrable on a compact interval r i=1 [a i , b i ] ⊂ R r . For a given compact interval s j =1 [c j , d j ] ⊂ R s , set T f s j =1

A multidimensional integration by parts formula for the Henstock-Kurzweil integral

“…(HK) The proof of Theorem 4.10 depends heavily on (11), which is also true for some other generalized Riemann integrals; more precisely, we have R e m a r k 4.11. Theorem 4.10 also holds if the Henstock-Kurzweil integral is replaced by any of the following generalized Riemann integrals:…”

Fubini Type Theorems for the strong McShane and strong Henstock-Kurzweil integrals

“…Henstock in [6] proved a Fubini-Tonelli type theorem for the Perron integral of real valued functions defined on two-dimensional compact intervals. Tuo-Yeong Lee in [17] and [18] proved several Fubini-Tonelli type theorems for the Henstock-Kurzweil integral of real valued functions defined on m-dimensional compact intervals in terms of the Henstock variational measures. In this paper, we will prove a Fubini type theorem for the strong Henstock-Kurzweil integral of Banach spaces valued functions defined on two-dimensional compact intervals, see Theorem 2.5.…”

A Fubini Theorem in Riesz spaces for the Kurzweil‐Henstock Integral

“…The Fubini-Tonelli Theorem for the Kurzweil-Henstock integral is well known when the functions involved are defined on bounded n-dimensional intervals, see for example [11,14,15,18,19,23] or [16] for unbounded intervals.…”