114 N. H. Bingham <strong>and</strong> A. J. OstaszewskiTheorem 11.5 (Nikodym’s Theorem – [Nik]; [Jay-Rog, p. 42]). The Baire sets of a spaceX are closed under the Souslin operation. Hence Souslin-F(X) sets are Baire.We promised examples of Baire sets; we can describe a hierarchy of them.Examples of Baire sets. By analogy with the projective hierarchy of sets (known alsoas the Luzin hierarchy – see [Kech], p. 313, which may be generated from the closed setsby iterating projection <strong>and</strong> complementation any finite number of times), we may formthe closely associated hierarchy of sets starting with the closed sets <strong>and</strong> iterating anyfinite number of times the Souslin operation S (following the notation of [Jay-Rog]) <strong>and</strong>complementation, denoted analogously by C say. Thus in a complete metric space oneobtains the family A of analytic sets, by complementation the family CA of co-analyticsets, then SCA which contains the previous two classes, <strong>and</strong> so on. By Nikodym’s theoremall these sets have the Baire property. One might call this the Souslin hierarchy.One may go further <strong>and</strong> form the smallest σ-algebra (with complementation allowed)closed under S <strong>and</strong> containing the closed sets; this contains the Souslin hierarchy (implicitthrough an iteration over the countable ordinals). Members of the latter family arereferred to as the C-sets – see Nowik <strong>and</strong> Reardon [NR].Definitions. 1. Say that a function f : X → Y between two <strong>topological</strong> spaces is H-Baire, for H a class of sets in Y, if f −1 (H) has the Baire property for each set H in H.Thus f is F(Y )-Baire if f −1 (F ) is Baire for all closed F in Y. Taking complements, sincef −1 (Y \H) = X\f −1 (H),f is F(Y )-Baire iff it is G(Y )-Baire, when we will simply say that f is Baire (‘f has theBaire property’ is the alternative usage).2. One must distinguish between functions that are F(Y )-Baire <strong>and</strong> those that lie in thesmallest family of functions closed under pointwise limits of sequences <strong>and</strong> containing thecontinuous functions (for a modern treatment see [Jay-Rog, Sect. 6]). We follow traditionin calling these last Baire-measurable (originally called by Lebesgue the analyticallyrepresentable functions, a term used in the context of metric spaces in [Kur-1, 2.31.IX,p. 392] ; cf. [Fos]).3. We will say that a function is Baire-continuous if it is continuous when restricted tosome co-meagre set. In the real line case <strong>and</strong> with the density topology, this is Denjoy’sapproximate continuity ([LMZ, p. 1]); recall ([Kech, 17.47]) that a set is (Lebesgue)measurable iff it has the Baire property under the density topology.The connections between these concepts are given in the theorems below. See thecited papers for proofs, <strong>and</strong> for the starting point, Baire’s Theorem on the points ofdiscontinuity of a Borel measurable function.
<strong>Normed</strong> <strong>groups</strong> 115Theorem 11.6 (Discontinuity Set Theorem – [Kur-1, p. 397] or [Kur-A]; [Ba1]; cf. [Ne,I.4]). (i) For f : X → Y Baire-measurable, with X, Y metric, the set of discontinuitypoints is meagre; in particular(ii) for f : X → Y Borel-measurable of class 1, with X, Y metric <strong>and</strong> Y separable, theset of discontinuity points is meagre.In regard to (ii) see [Han-71, Th. 10] for the non-separable case. The following theorem,from recent literature, usefully overlaps with the last result.Theorem 11.7 (Banach-Neeb Theorem – [Ban-T], Th. 4 pg. 35 <strong>and</strong> Vol I p. 206; [Ne] (i)I. 6 (ii) I.4). (i) A Borel-measurable f : X → Y with X, Y metric <strong>and</strong> Y separable <strong>and</strong>arcwise connected is Baire-measurable; <strong>and</strong>(ii) a Baire-measurable f : X → Y with X a Baire space <strong>and</strong> Y metric is Bairecontinuous.Remark. In fact Banach shows that a Baire-measurable function is Baire-continuouson each perfect set ([Ban-T, Vol. II p. 206]). In (i) if X, Y are completely metrizable,<strong>topological</strong> <strong>groups</strong> <strong>and</strong> f is a homomorphism, Neeb’s assumption that Y is arcwise connectedbecomes unecessary, since, as Pestov [Pes] remarks, the arcwise connectednessmay be dropped by referring to a result of Hartman <strong>and</strong> Mycielski [HM] that a separablemetrizable group embeds as a subgroup of an arcwise connected separable metrizablegroup.Theorem 11.8 (Baire Continuity Theorem). A Baire function f : X → Y is Bairecontinuousin the following cases:(i) Baire condition (see e.g. [HJ, Th. 2.2.10 p. 346]): Y is a second-countable space;(ii) Emeryk-Frankiewicz-Kulpa ([EFK]): X is Čech-complete <strong>and</strong> Y has a base of cardinalitynot exceeding the continuum;(iii) Pol condition ([Pol]): f is Borel, X is Borelian-K <strong>and</strong> Y is metrizable <strong>and</strong> of nonmeasurablecardinality;(iv) Hansell condition ([Han-71]): f is σ-discrete <strong>and</strong> Y is metric.We will say that the pair (X, Y ) enables Baire continuity if the spaces X, Y satisfyeither of the two conditions (i) or (ii) above. In the applications below Y is usually theadditive group of reals R, so satisfies (i). Building on [EFK], Fremlin ([Frem, Section 10])characterizes a space X such that every Baire function f : X → Y is Baire-continuousfor all metric Y in the language of ‘measurable spaces with negligibles’; reference thereis made to disjoint families of negligible sets all of whose subfamilies have a measurableunion. For a discussion of discontinuous homomorphisms, especially counterexamples onC(X) with X compact (e.g. employing Stone-Čech compactifications, X = βN\N ), see[Dal, Section 9].
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N. H. BINGHAM and A. J. OSTASZEWSKI
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Normed groups 3ContentsContents . .
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1. IntroductionGroup-norms, which b
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Normed groups 3Topological complete
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Normed groups 5abelian group has se
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Normed groups 74 (Topological permu
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Normed groups 9The following result
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Normed groups 11Corollary 2.4. For
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Normed groups 13More generally, for
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Normed groups 15definitions, our pr
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Normed groups 17so that fg is in th
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Normed groups 19(iii) The ¯d H -to
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Normed groups 21so‖αβ‖ ≤
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Normed groups 23Remark. Note that,
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Normed groups 25shows that [z n , y
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Normed groups 27Denoting this commo
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Normed groups 29Theorem 3.4 (Equiva
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Normed groups 31argument as again p
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Normed groups 33(ii) For α ∈ H u
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Normed groups 35Definition. A group
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Normed groups 37We now give an expl
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Normed groups 39Theorem 3.19 (Abeli
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Normed groups 412. Further recall t
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Normed groups 43Theorem 3.22 (Lipsc
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Normed groups 45Proof. Z γ = G (cf
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Normed groups 47Theorem 3.30. Let G
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Normed groups 49Remark. On the matt
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Normed groups 51As for the conclusi
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Normed groups 53By (C-adm), we may
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Normed groups 55equipped with an in
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Normed groups 57Proof. To apply Th.
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Normed groups 59Definition. A point
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Normed groups 61Proposition 3.46 (M
- Page 67 and 68: Normed groups 63Thus ω δ (s) ≤
- Page 69 and 70: Normed groups 65Remark. In the penu
- Page 71 and 72: Normed groups 67The result confirms
- Page 73 and 74: Normed groups 69Proof. By the Baire
- Page 75 and 76: Normed groups 715. Generic Dichotom
- Page 77 and 78: Normed groups 73Returning to the cr
- Page 79 and 80: Normed groups 75Examples. Here are
- Page 81 and 82: Normed groups 77cf. [Eng, 4.3.23].)
- Page 83 and 84: Normed groups 79Remarks. 1. See [Fo
- Page 85 and 86: Normed groups 81Theorem 6.1 (Catego
- Page 87 and 88: Normed groups 83is continuous at th
- Page 89 and 90: Normed groups 85compact. Evidently,
- Page 91 and 92: Normed groups 87j ∈ ω} which enu
- Page 93 and 94: Normed groups 89The result below ge
- Page 95 and 96: Normed groups 91left-shift, not in
- Page 97 and 98: Normed groups 93As a corollary of t
- Page 99 and 100: Normed groups 953. For X a normed g
- Page 101 and 102: Normed groups 97Proof. Note that‖
- Page 103 and 104: Normed groups 99Taking h(x) := ‖
- Page 105 and 106: Normed groups 1019. The Semigroup T
- Page 107 and 108: Normed groups 103Theorem 9.5 (Semig
- Page 109 and 110: Normed groups 105By the Category Em
- Page 111 and 112: Normed groups 107Proof. Say f is bo
- Page 113 and 114: Normed groups 109Thus G is locally
- Page 115 and 116: Normed groups 111Theorem 10.10 (Bar
- Page 117: Normed groups 113K-analyticity was
- Page 121 and 122: Normed groups 117restricted to X\M
- Page 123 and 124: Normed groups 119groups need not be
- Page 125 and 126: Normed groups 121Proof. In the meas
- Page 127 and 128: Normed groups 123Hence, as t i n
- Page 129 and 130: Normed groups 125The corresponding
- Page 131 and 132: Normed groups 127(t, x) ✛✻Φ T
- Page 133 and 134: Normed groups 129Fix s. Since s is
- Page 135 and 136: Normed groups 131Hence,‖x‖ −
- Page 137 and 138: Normed groups 133converging to the
- Page 139 and 140: Normed groups 135Definition. Let {
- Page 141 and 142: Normed groups 137However, whilst th
- Page 143 and 144: Normed groups 139embeddable, 14enab
- Page 145 and 146: Normed groups 141Bibliography[AL]J.
- Page 147 and 148: Normed groups 143Series 378, 2010.[
- Page 149 and 150: Normed groups 145abelian groups, Ma
- Page 151 and 152: Normed groups 147[Kak] S. Kakutani,
- Page 153 and 154: Normed groups 149fields. I. Basic p
- Page 155: Normed groups 151[So]R. M. Solovay,