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On Denjoy type extensions of the Pettis integral

Kirill Naralenkov (2010)

Czechoslovak Mathematical Journal

In this paper two Denjoy type extensions of the Pettis integral are defined and studied. These integrals are shown to extend the Pettis integral in a natural way analogous to that in which the Denjoy integrals extend the Lebesgue integral for real-valued functions. The connection between some Denjoy type extensions of the Pettis integral is examined.

On Denjoy-Dunford and Denjoy-Pettis integrals

José Gámez, José Mendoza (1998)

Studia Mathematica

The two main results of this paper are the following: (a) If X is a Banach space and f : [a,b] → X is a function such that x*f is Denjoy integrable for all x* ∈ X*, then f is Denjoy-Dunford integrable, and (b) There exists a Dunford integrable function f : [ a , b ] c 0 which is not Pettis integrable on any subinterval in [a,b], while ʃ J f belongs to c 0 for every subinterval J in [a,b]. These results provide answers to two open problems left by R. A. Gordon in [4]. Some other questions in connection with Denjoy-Dundord...

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