Mat 201b Recap
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4. Let (Ω,Σ, μ) be a measure space. If f : Ω → R, then we say that f is a measurable function if for every t ∈ R, the level set: Sf (t) = {x ∈ Ω|f(x) > t} is measurable. f : Ω→ C is measurable if and only if both Re(f) and Im(f) are measurable. 5. A real function f : Ω → R is lower semicontinuous on Ω if Sf (t) is open, and is upper semicontinuous if {x ∈ Ω|f(x) < t} is open. Equivalently, f is lower semicontinuous if ∀ > 0, ∃ open ball B ⊂ Ω containing x with f(y) ≥ f(x)− for every y ∈ B (equivalently, f(x) ≤ lim inf n→∞ f(xn) for all xn → x), and is upper semicontinuous if f(y) ≤ f(x) + (equivalently, f(x) ≥ lim sup n→∞ f(xn) for all xn → x). 6. Consider measure space (R,B, μ), and f is a Borel measurable function. Let Ω = {A ⊂ R|A is open and f(x) = 0 for μ-a.e. x ∈ A}, then the essential support of f is ( ⋃ A∈Ω A) , which is a closed set. 7. Let (Ω,Σ, μ) be a measure space, and f : Ω → R be measurable, then the integral of f is ∫
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Mat 215a Recap
(a) D: α0 = 1, α1 = 1, α2 = 1. χ(D ) = 1. (b) S: α0 = 1, α1 = 0, α2 = 1. χ(S ) = 2. (c) Möbius band: α0 = 1, α1 = 2, α2 = 1. χ(M) = 0. (d) Mg of genus g: α0 = 1, α1 = 2g, α2 = 1. χ(Mg) = 2− 2g. 6. RP n = (R − {0})/{z ∼ λz} = S/{z ∼ −z} = S/Z2 RP n = σ ∪ RP n−1 implies RP n = σ ∪ σn−1 ∪ · · · ∪ σ χ(RP ) = 1 for n even and 0 for n odd. 7. CP n = (C − {0})/{z ∼ λz} = S/{z ∼ λz||λ| = 1} CP n = σ ∪ ...
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