By S. Zaidman
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Extra resources for Almost-periodic functions in abstract spaces
Proof. 9, each function fn is a s-discrete function of the ﬁrst Borel class. Hence the limit function is of the ﬁrst Borel class. 5, the limit function is s-discrete. 11 Let f be a function from one metric space X to another Y. If f has a s-discrete base of closed sets, then the set of points of discontinuity of f is an F s -set of the ﬁrst Baire category in X. A DISJOINT DISCRETELY s-DECOMPOSABLE FAMILY OF F s -SETS 31 Proof. Let B ¼ fBu : u [ Qg be a closed s-discrete base for f with Q¼ 1 [ QðnÞ; n¼1 and, with each family B n ¼ fBu : u [ QðnÞg; a discrete family of closed sets.
Then x belongs to none of the sets Ug with g – gÃ and so, for each pair j, k with j þ k # i þ 1, the point x belongs to no set in the discrete family fJgðj;kÞ : g [ G; g – gÃ g and we can choose a neighborhood N ðj;kÞ of x that meets none of these sets. Thus \ N ðj;kÞ N¼ jþk#iþ1 is a neighborhood of x that meets none of the sets A DISJOINT DISCRETELY s-DECOMPOSABLE FAMILY OF F s -SETS FgðiÞ ¼ [ Jgðj;kÞ ; 23 g [ G; g – gÃ : jþk#iþ1 Thus N meets at most one of the sets fFgðiÞ : g [ Gg. Similarly, if x belongs to no set Ug , g [ G, then N can be chosen to meet no set of the family fFgðiÞ : g [ Gg.
1(c)]. 3 Let X be a metric space and let K Ã be a convex weakÃ closed set in the dual Y Ã of a Banach space Y (perhaps K Ã ¼ Y Ã ). Suppose that ðK Ã ; weakÃ Þ is s-fragmented by the norm using weakÃ closed sets. If F is any upper semi-continuous set-valued function from X to ðK Ã ; weakÃ Þ taking only nonempty weakÃ compact values, then F has a selector f : X ! ðK Ã ; normÞ that is of the ﬁrst Baire class and has an F s -set of the ﬁrst Baire category as its set of points of discontinuity. Note that the dual space Y Ã has the Radon–Nikody´m property if and only if the unit ball ðBÃ ; weakÃ ) of Y Ã is fragmented by the norm.