MR3191427 Naralenkov, Kirill M., A Lusin type measurability property for vector-valued functions. J. Math. Anal. Appl. 417 (2014), no. 1, 293307. 28A20

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Abstract

In the paper under review the author introduces the notion of Riemann measurability for vector-valuedfunctions, generalizing the classical Lusin condition, which is equivalent to the Lebesgue measurabilityfor real valued functions. Let X be a Banach space, let f : [a; b] ! X and let E be a measurable subset of [a; b]. The functionf is said to be Riemann measurable on E if for each " > 0 there exist a closed set F E with (E n F) < 0 (where is the Lebesgue measure) and a positive number such thatk XKk=1ff(tk) ?? f(t0k)g (Ik)k < "whenever fIkgKk=1 is a nite collection of pairwise non-overlapping intervals with max1 k K (Ik) < and tk; t0k 2 IkTF.The Riemann measurability is more relevant to Riemann type integration theory, such as those ofMcShane and Henstock, rather than the classical notion of Bochner or scalar measurability. In par-ticular the author studies the relationship between the Riemann measurability and the M and the Hintegrals that are obtained if we assume that the gauge in the de nitions of McShane and Henstockintegral can be chosen Lebesgue measurable.The main results are the following If f : [a; b] ! X is H-integrable on a measurable subset E of [a; b], then f is Riemann measurableon E. If f : [a; b] ! X is both bounded and Riemann measurable on a measurable subset E of [a; b], thenf is M-integrable on E. If f : [a; b] ! X is both Riemann measurable and McShane (Henstock) integrable on a measurablesubset E of [a; b], then f is M-integrable (H-integrable) on E. Suppose X separable. If f : [a; b] ! X is McShane (Henstock) integrable, then f is M-integrable(H-integrable.)The author concludes the paper with the following open problem: for which families of non-separableBanach spaces does the McShane (or even the Pettis) integrability imply Riemann measurability?Reviewed by (L. Di Piazza)
Lingua originaleEnglish
Numero di pagine1
RivistaElectronic Mathematical Reviews (eMR) Sections
Volume2014 MR3191427
Stato di pubblicazionePublished - 2014

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