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- W19790419 endingPage "586" @default.
- W19790419 startingPage "531" @default.
- W19790419 abstract "Until 1968, there has been no significant progress in the theory of the Pettis integral since Pettis's (1938) original paper. The Pettis integral was known to be countably additive and absolutely continuous. Due to Pettis (1938) it was also known that the space of Pettis integrable functions defined on the unit interval, integrated with respect to the Lebesgue measure may be non-complete in the semivariation norm. There were also known some facts concerning Pettis integrability of strongly measurable functions and a few examples of non-strongly measurable but Pettis integrable functions but almost nothing more. Then, Rybakov (1968) proved that the Pettis integral is of σ-finite variation. In 1975, Thomas presented some negative aspects of Pettis integrability. He proved that the space of Pettis integrable functions with respect to a nonatomic measure in never complete if the range space is infinite dimensional and that for each infinite dimensional Banach space X there exists an X -valued function on [0, 1] 2 such that not almost all its section are Pettis integrable on [0, 1]. Thomas also suggested that probably one could find a Pettis integrable function f on [0, 1] such that lim h → 0 ‖ 1 h ∫ s s + h f ( t ) dt ‖ = ∞ on a set of positive Lebesgue measure." @default.
- W19790419 created "2016-06-24" @default.
- W19790419 creator A5022646520 @default.
- W19790419 date "2002-01-01" @default.
- W19790419 modified "2023-09-23" @default.
- W19790419 title "Pettis Integral" @default.
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- W19790419 doi "https://doi.org/10.1016/b978-044450263-6/50013-0" @default.
- W19790419 hasPublicationYear "2002" @default.
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