Banach Lattices and the Dual of C(X)

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$MATH 5b; Solutions to Homework Set 1 January 2009 1. As K is a ...$

4 NOTES FOR MA 110A we find that For λ > 0 and real, and ℓ+(g + h) = ℓ+(g+ + h+) − ℓ+(g− + h−) = ℓ+(g+) + ℓ+(h+) − ℓ−(g−) − ℓ+(h−) = ℓ+(g) + ℓ+(h) ℓ+(λf) = ℓ+(λf+) − ℓ+(λf−) = λℓ+(f) ℓ+(−f) = ℓ+(f−) − ℓ+(f+) = −ℓ+(f) Therefore, ℓ+ is a linear functional. Clearly, so and ℓ = ℓ+ − ℓ−. Note that ℓ+(f) ≥ ℓ(f) ℓ− ≡ ℓ+ − ℓ ≥ 0 ℓ−(f) = sup(ℓ(g − f) | 0 ≤ g ≤ f) = sup(−ℓ(h) | 0 ≤ h ≤ f) (2.4) by taking h = f − g. Suppose ℓ = k+−k−. Then 0 ≤ g ≤ f implies ℓ(g) = k+(g)−k−(g) ≤ k+(g) ≤ k+(f), so taking sups, ℓ+ ≤ k+. Using (2.4), we get ℓ− ≤ k−. Finally, given f ≥ 0 and g, h so 0 ≤ g ≤ f, 0 ≤ h ≤ f, we have −f ≤ g − h ≤ f, so |g − h| ≤ f and so by the Banach lattice property g − h ≤ f. Thus, |ℓ(g) − ℓ(h)| = |ℓ(g − h)| ≤ ℓ g − h ≤ ℓ f Taking the sup of ℓ(g) over all g and −ℓ(h) over all h, we find that (2.2) holds. Corollary. If ℓ ∈ C(X) ∗ , then ℓ = ℓ+ − ℓ− with ℓ± ≥ 0 and ℓ+ + ℓ = ℓ (2.5)
NOTES FOR MA 110A 5 Proof. We only need to check (2.5). But ℓ± = ℓ±(1), so ℓ+ + ℓ− = ℓ+(1) + ℓ−(1) ≤ ℓ by (2.2) with f = 1. Since ℓ = ℓ+ − ℓ−, ℓ ≤ ℓ+ + ℓ− proving (2.5). A signed Baire measure on X is an assignment of a real number, µ(A), to each Baire set so that there is C < ∞ obeying if X = A1 ∪ · · · ∪ An and Ai ∩ Aj = ∅, then n |µ(Aj)| ≤ C (2.6) j=1 and for any countable family of disjoint sets {Aj} n j=1 , we have µ(∪Aj) = µ(Aj) (the sum converges absolutely by (2.6)). Given such a measure, as usual we can form a Stieltjes-type functional ℓ on C(X) with |ℓ(f)| ≤ f∞ so ℓ = ℓ+ − ℓ− and µ = µ+ − µ−. We set |µ| = µ+ + µ−. Conversely, any µ+ − µ− defines a signed measure. We thus have the Riesz–Markov theorem for signed measures which says there is a oneone correspondence between signed measures and elements of C(X) ∗ . 3. Hahn and Jordan Decompositions As we have seen, every signed measure µ induces an ℓ which can be written as µ+ − µ− where µ± obeys µ+ + µ− = µ and if µ = ν+ − ν− with ν± ≥ 0, then µ± ≤ ν±. Here we will prove two famous theorems: Hahn Decomposition Theorem. There exists a Baire set, A, so that for all C, µ+(C) = µ(A ∩ C) (3.1) µ−(C) = −µ((X \ A) ∩ C) (3.2) µ = µ(A) + µ(X \ A) (3.3) Jordan Decomposition Theorem. µ+ and µ− are mutually singular. Recall |µ| is the positive measure µ+ + µ−. We will deduce the two theorems above from Theorem. There exists a Baire function, g, with |g(x)| = 1 for d|µ| a.e. x so that f dµ = fg d|µ| (3.4)
Page 1 and 2: Ma 110a B. Simon Fall 2006 GENERAL
Page 3: (ii) For any f ≥ 0, we have NOTES
Page 7: For f ∈ C(X), let ν+(f) = NOTES

Banach Lattices and the Dual of C(X)

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