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 ∏i=1r[ai,bi]⊂Rr. For a given compact interval ∏j=1s[cj,dj]⊂Rs, setTf(∏j=1s[cj,dj]):={g:f⊗g∈HK(∏i=1r[ai,bi]×∏j=1s[cj,dj])}. We prove that if g∈Tf(∏j=1s[cj,dj]) and ν   is a finite signed Borel measure on ∏j=1s[cj,dj), then the function (y1,…,ys)↦g(y1,…,ys)ν(∏j=1s[cj,yj)) belongs to Tf(∏j=1s[cj,dj]). Moreover, this result cannot be improved.