CHANGE OF VARIABLE THEOREMS FOR THE KH INTEGRAL

摘要

Let f: [a,b] (is contained in) R → ε and φ : [a,b] → F, where (ε,F,G) is a Banach space triple, a) We prove that if φ is continuous [c, d] → [a, b] and f o ψ ? dφ o ψ is Kurzweil or Henstock variationally integrable, then so is f ? dφ and fulfills the well known change of variable formula. It follows that if ψ is an indefinite Henstock integral and if f o ψψ' dx is K-H integrable, then so is fdx and the change of variable formula applies, b) We produce several versions of the converse of a), that is, we give necessary and sufficient conditions in order that with ψ as above, the integrability of f ? dφ implies that of f o ψ ? dφ o ψ and the change of variable formula.