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- Title
CHANGE OF VARIABLE THEOREMS FOR THE KH INTEGRAL.
- Authors
Bensimhoun, Michael
- Abstract
Let ƒ : [a; b] ⊆ ℝ̄ → E and ' : [a; b] → F, where (E;F; G) is a Banach space triple. a) We prove that if φ is continuous [c; d] → [a; b] and ƒ ⚬ ψ · dφ ⚬ ψ is Kurzweil or Henstock variationally integrable, then so is ƒ · dφ and fulfills the well known change of variable formula. It follows that if ψ is an indefinite Henstock integral and if f ⚬ ψ ψ' dx is K-H integrable, then so is ƒdx 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 ƒ · dφ implies that of ƒ ⚬ ψ · dψ ⚬ ψ and the change of variable formula.
- Subjects
HENSTOCK-Kurzweil integral; STIELTJES integrals; FUNCTIONAL integration; RIEMANN integral; GENERALIZED integrals; BANACH spaces; REAL variables; COMPLEX variables
- Publication
Real Analysis Exchange, 2010, Vol 35, Issue 1, p167
- ISSN
0147-1937
- Publication type
Article
- DOI
10.14321/realanalexch.35.1.0167