The Dual of L∞(X,L,λ), Finitely Additive Measures and Weak Convergence A Primer

Preface......Page 6
Contents......Page 9
1 Introduction......Page 11
2.1 Partial Ordering and Zorn's Lemma......Page 16
2.2 Algebras and σ-Algebras......Page 17
2.3 Measurable Sets and Measurable Functions......Page 18
2.4 Measures and Real Measures......Page 19
2.5 Splitting Families of Measurable Sets......Page 23
2.6 Integration......Page 24
2.7 Function Spaces......Page 26
2.8 Dual Spaces and Measures......Page 28
2.9 Functional Analysis......Page 29
2.10 Point Set Topology......Page 32
3 Linfty and Its Dual......Page 36
4.1 Definition, Notation and Basic Properties......Page 39
4.2 Purely Finitely Additive Measures......Page 44
4.3 Canonical Decomposition: ba(mathcalL) = Σ(mathcalL) oplusΠ(mathcalL)......Page 46
4.4 Linfty*(X, mathcalL, λ)......Page 47
5 mathfrakG: 0–1 Finitely Additive Measures......Page 48
5.1 mathfrakG and Ultrafilters......Page 49
5.2 mathfrakG and the λ-Finite Intersection Property......Page 51
6.1 The Integral......Page 54
6.2 Yosida–Hewitt Representation: Proof of Theorem3.1......Page 57
6.3 Integration with Respect to ωinmathfrakG......Page 58
6.5 Integrating u inellinfty(mathbbN) with Respect to mathfrakG......Page 59
6.6 The Valadier–Hensgen Example......Page 61
7.1 The Space (mathfrakG,τ)......Page 64
7.2 Linfty(X,mathcalL,λ) and C(mathfrakG, τ) Isometrically Isomorphic......Page 65
7.3 Properties of mathfrakG and τ......Page 66
7.4 mathfrakG and the Weak* Topology on Linfty*(X, mathcalL, λ)......Page 70
7.5 mathfrakG as Extreme Points......Page 71
8.1 Weakly Convergent Sequences......Page 74
8.2 Pointwise Characterisation......Page 75
8.3 Applications of Theorem 8.7......Page 79
9.1 Localising mathfrakG......Page 83
9.2 Localising Weak Convergence......Page 86
9.3 Fine Structure at x0 of u inLinfty(X, mathcalB, λ)......Page 87
9.4 A Localised Range from Complex Function Theory......Page 90
10.1 (mathfrakG, τ) Versus Linfty(X, mathcalL, λ)......Page 93
10.2 Restriction to C0(X, ) of Elements of Linfty*(X, mathcalB,λ)......Page 95
BookmarkTitle:......Page 100
Index......Page 102