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

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[ai, bi] ⊂ Rr . For a given compact interval s j=1[cj , dj] ⊂ Rs, set Tf s j=1 [cj , dj ] := g: f ⊗g ∈HK r i=1 [ai, bi] × s j=1 [cj , dj ] . We prove that if g ∈ Tf ( s j=1[cj , dj ]) and ν is a finite signed Borel measure on s j=1[cj , dj ), then the function (y1, . . . , ys ) →g(y1, . . . , ys)ν( s j=1[cj , yj )) belongs to Tf ( s j=1[cj , dj ]). Moreover, this result cannot be improved. © 2005 Elsevier Inc. All rights reserved