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RESEARCH PRODUCT

MR2569913: Rodríguez, José. Some examples in vector integration. Bull. Aust. Math. Soc. 80 (2009), no. 3, 384–392. (Reviewer: Luisa Di Piazza),

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The paper deals with some classical examples in vector integration due to Phillips, Hagler and Talagrand, revisited from the point of view of the Birkhoff and McShane integrals. More precisely, the author considers: - Phillips' example of a Pettis integrable function f which is not Birkhoff integrable [R. S. Phillips, Trans. Amer. Math. Soc. 47 (1940), 114--145; MR0002707 (2,103c)]. It is proved here that f is universally McShane integrable. - Hagler's example of a scalarly measurable l∞-valued function g which is not strongly measurable. The function g is proved to be universally Birkhoff integrable. - Talagrand's example of a bounded Pettis integrable function φ having no conditional expectation [M. Talagrand, Mem. Amer. Math. Soc. 51 (1984), no. 307, ix+224 pp.; MR0756174 (86j:46042)]. Here the author shows that φ is also Birkhoff integrable, giving a negative answer to the question whether conditional expectations exist within the Birkhoff theory. Some interesting open problems are also stated. Reviewed by Luisa Di Piazza