MR2657294 (2011h:28021) Bensimhoun, Michael Change of variable theorems for the KH integral. Real Anal. Exchange 35 (2010), no. 1, 167–194. (Reviewer: Luisa Di Piazza), 28B05 (26A42 46G10)

From Reviews: 0 MR2657294 (2011h:28021) Bensimhoun, Michael(IL-HEBR) Change of variable theorems for the KH integral. (English summary) Real Anal. Exchange 35 (2010), no. 1, 167–194. 28B05 (26A42 46G10) PDF Clipboard Journal Article Make Link Let $({\scr E},{\scr F}, {\scr G})$(E,F,G) be a Banach triple and let $f\colon [a,b] \subset \overline{\Bbb R} \rightarrow {\scr E}$f:[a,b]⊂R−−→E, $\varphi \colon [a,b] \rightarrow {\scr F}$φ:[a,b]→F and $\psi\colon [c,d] \subset \overline{\Bbb R} \rightarrow [a,b]$ψ:[c,d]⊂R−−→[a,b] be given. The problem of change of variables in an integral consists in finding the best conditions under which the equality $$ \int_c^d f \circ \psi \cdot d(\varphi \circ \psi) = \int_{\psi(c)}^{\psi(d)} f \cdot d\varphi\tag1 $$∫dcf∘ψ⋅d(φ∘ψ)=∫ψ(d)ψ(c)f⋅dφ(1) holds, when one of these two integrals exists. Here the context is that of the Kurzweil-Henstock-Stieltjes integral. The main result is Theorem 6.1: Assume that $\psi$ψ is continuous. If $f \circ \psi \cdot d(\varphi \circ \psi)$f∘ψ⋅d(φ∘ψ) is integrable in $[c,d]$[c,d], then $f \cdot d\varphi$f⋅dφ is integrable in $\psi([c, d])$ψ([c,d]) and equality (1) holds. Furthermore, if $f \circ \psi \cdot d(\varphi \circ \psi)$f∘ψ⋅d(φ∘ψ) is absolutely integrable in $[c,d]$[c,d], then $f \cdot d\varphi$f⋅dφ is absolutely integrable in $\psi([c, d])$ψ([c,d]), with $$ \int_c^d \|f \circ \psi \cdot d(\varphi \circ \psi)\| = \int_{\psi(c)}^{\psi(d)} \|f \cdot d\varphi\|. $$∫dc∥f∘ψ⋅d(φ∘ψ)∥=∫ψ(d)ψ(c)∥f⋅dφ∥. Without very specific additional conditions on $f$f, the continuity of $\psi$ψ is essential in order to get relation (1). As a corollary of Theorem 6.1, the author obtains, with an alternative proof, a formula for change of variables in [S. Leader, Real Anal. Exchange 29 (2003/04), no. 2, 905--920; MR2083825 (2005f:26023)], for the case $\varphi={\rm Id}$φ=Id and $\psi$ψ real-valued of bounded variation. In the second part of the paper, necessary and sufficient conditions are given in order that the integrability of $f \cdot d\varphi$f⋅dφ implies that of $f \circ \psi \cdot d(\varphi \circ \psi)$f∘ψ⋅d(φ∘ψ) and the change of variable formula. Reviewed by Luisa Di Piazza