92 N. H. Bingham <strong>and</strong> A. J. OstaszewskiWe now see that the preceding theorem is applicable to a Haar measure on a locallycompact group X by reference to the following result. Here bounded means pre-compact(covered by a compact set).Theorem 7.3 (Haar measure density theorem – [Mue]; cf. [Hal-M, p. 268]). Let A be aσ-bounded subset <strong>and</strong> µ a left-invariant Haar measure of a locally compact <strong>topological</strong>group X. Then there exists a sequence U n of bounded measurable neigbourhoods of e Xsuch that m ∗ (A ∩ U n x)/m ∗ (U n x) → 1 for almost all x out of a measurable cover of A.Corollary 7.4. In the setting of Theorem 6.4 with A of positive, totally-finite Haarmeasure, let (A, S A , m A ) be the induced probability subspace of X with m A (T ) = m(S ∩A)/m(A) for T = S ∩ A ∈ S A . Then the density theorem holds in A.We now offer a generalization of a result from [BOst-LBII]; cf. Theorem 6.2.Theorem 7.5 (Second Verification Theorem for weak category convergence). Let X bea locally compact <strong>topological</strong> group with left-invariant Haar measure m. Let V be m-measurable <strong>and</strong> non-null. For any null sequence {z n } → e <strong>and</strong> each k ∈ ω,H k = ⋂ n≥k V \(V · z n) is of m-measure zero, so meagre in the d-topology.That is, the sequence h n (x) := xzn−1(wcc)satisfies the weak category convergence conditionProof. Suppose otherwise. We write V z n for V · z n , etc. Now, for some k, m(H k ) > 0.Write H for H k . Since H ⊆ V, we have, for n ≥ k, that ∅ = H ∩ h −1n (V ) = H ∩ (V z n )<strong>and</strong> so a fortiori ∅ = H ∩ (Hz n ). Let u be a metric density point of H. Thus, for somebounded (Borel) neighbourhood U ν u we haveFix ν <strong>and</strong> putm[H ∩ U ν u] > 3 4 m[U νu].δ = m[U ν u].Let E = H ∩ U ν u. For any z n , we have m[(Ez n ) ∩ U ν uz n ] = m[E] > 3 4δ. By TheoremA of [Hal-M, p. 266], for all large enough n, we havem(U ν u△U ν uz n ) < δ/4.Hence, for all n large enough we have m[(Ez n )\U ν u] ≤ δ/4. Put F = (Ez n ) ∩ U ν u; thenm[F ] > δ/2.But δ ≥ m[E ∪ F ] = m[E] + m[F ] − m[E ∩ F ] ≥ 3 4 δ + 1 2δ − m[E ∩ F ]. Som[H ∩ (Hz n )] ≥ m[E ∩ F ] ≥ 1 4 δ,contradicting ∅ = H ∩ (Hz n ). This establishes the claim.
<strong>Normed</strong> <strong>groups</strong> 93As a corollary of the Category Embedding Theorem, Theorem 6.5 <strong>and</strong> its Corollarynow yield the following result (compare also Th. 10.11).Theorem 7.6 (First Generalized Kestelman-Borwein Ditor Theorem – Measurable Case).Let X be a normed locally compact group, {z n } → e X be a null sequence in X. If T isHaar measurable <strong>and</strong> non-null (resp. Baire <strong>and</strong> non-meagre), then for generically allt ∈ T there is an infinite set M t such that{tz m : m ∈ M t } ⊆ T.This theorem in turn yields an important conclusion.Theorem 7.7 (Kodaira’s Theorem – [Kod] Corollary to Satz 18. p. 98, cf. [Com, Th.4.17 p.1182]). Let X be a normed locally compact group <strong>and</strong> f : X → Y a homorphisminto a separable normed group Y . Then f is Haar-measurable iff f is Baire under thedensity topology iff f is continuous under the norm topology.Proof. Suppose that f is measurable. Then under the d-topology f is a Baire function.Hence by the classical Baire Continuity Theorem (see, e.g. Section 11 below, especiallyTh.11.8), since Y is second-countable, f is continuous on some co-meagre set T. Nowsuppose that f is not continuous at e X . Hence, for some ε > 0 <strong>and</strong> some z n → z 0 = e X (inthe sense of the norm on X), we have ‖f(z n )‖ > ε, for all n. By the Kestelman-Borwein-Ditor Theorem (Th. 6.1), there is t ∈ T <strong>and</strong> an infinite M t such that tz n → t = tz 0 ∈ T.Hence, for n in M t , we havei.e. f(z n ) → e Y , a contradiction.f(t)f(z n ) = f(tz n ) → f(tz 0 ) = f(t),Remark. 1. Comfort [Com, Th. 4.17] proves this result for both X <strong>and</strong> Y locally compact,with the hypothesis that Y is σ-compact <strong>and</strong> f measurable with respect to the twoHaar measures on X <strong>and</strong> Y . That proof employs Steinhaus’ Theorem <strong>and</strong> the Weil topology.(Under the density topology, Y will not be second-countable.) When Y is metrizablethis implies that Y is separable; of course if f is a continuous surjection, Y will be locallycompact (cf.[Eng, Th.3.1.10], [Kel, Ch. V Th. 8]).2. The theorem reduces measurability to the Baire property <strong>and</strong> in so doing resolves along-st<strong>and</strong>ing issue in the foundations of regular variation; hitherto the theory was establishedon two alternative foundations employing either measurable functions or Bairefunctions for its scope, with historical preference for measurable functions in connectionwith integration. We refer to [BGT] for an exposition of the theory, which characterizesregularly varying functions of either type by a reduction to an underlying homomorphismof the corresponding type relying on its continuity, <strong>and</strong> then represents either type byvery well-behaved functions. Kodaira’s Theorem shows that the broader <strong>topological</strong> class
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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
- Page 45 and 46: Normed groups 412. Further recall t
- Page 47 and 48: Normed groups 43Theorem 3.22 (Lipsc
- Page 49 and 50: Normed groups 45Proof. Z γ = G (cf
- Page 51 and 52: Normed groups 47Theorem 3.30. Let G
- Page 53 and 54: Normed groups 49Remark. On the matt
- Page 55 and 56: Normed groups 51As for the conclusi
- Page 57 and 58: Normed groups 53By (C-adm), we may
- Page 59 and 60: Normed groups 55equipped with an in
- Page 61 and 62: Normed groups 57Proof. To apply Th.
- Page 63 and 64: Normed groups 59Definition. A point
- Page 65 and 66: 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: Normed groups 91left-shift, not in
- 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 and 118: Normed groups 113K-analyticity was
- Page 119 and 120: Normed groups 115Theorem 11.6 (Disc
- 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.
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Normed groups 143Series 378, 2010.[
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Normed groups 145abelian groups, Ma
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Normed groups 147[Kak] S. Kakutani,
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Normed groups 149fields. I. Basic p
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Normed groups 151[So]R. M. Solovay,