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The Denjoy extension of the Riemann and McShane integrals

Jae Myung Park — 2000

Czechoslovak Mathematical Journal

In this paper we study the Denjoy-Riemann and Denjoy-McShane integrals of functions mapping an interval a , b into a Banach space X . It is shown that a Denjoy-Bochner integrable function on a , b is Denjoy-Riemann integrable on a , b , that a Denjoy-Riemann integrable function on a , b is Denjoy-McShane integrable on a , b and that a Denjoy-McShane integrable function on a , b is Denjoy-Pettis integrable on a , b . In addition, it is shown that for spaces that do not contain a copy of c 0 , a measurable Denjoy-McShane integrable...

The M α and C -integrals

Jae Myung Park;  Hyung Won Ryu;  Hoe Kyoung Lee;  Deuk Ho Lee — 2012

Czechoslovak Mathematical Journal

In this paper, we define the M α -integral of real-valued functions defined on an interval [ a , b ] and investigate important properties of the M α -integral. In particular, we show that a function f : [ a , b ] → R is M α -integrable on [ a , b ] if and only if there exists an A C G α function F such that F ' = f almost everywhere on [ a , b ] . It can be seen easily that every McShane integrable function on [ a , b ] is M α -integrable and every M α -integrable function on [ a , b ] is Henstock integrable. In addition, we show that the M α -integral is equivalent to the C -integral....

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