推 jimlucky :謝謝~我看看~^^ 02/27 10:47
※ 引述《jimlucky (......)》之銘言：
: V is the collection of all finite signed measures on R^n with
: weak-star topology (generated by the family of semi norm < ,f> where
: <m,f>=∫fdm and f is continuous with compact support.)
: Show that V is separable.
: 最近讀東西遇到的問題，請大大幫忙了~謝謝
引進一個概念叫regularization
對於一個ε>0,
可以找到一個continuous function ρ_ε(x), support掉在ε的球裡面,
而且其Lebesgue積分=1
假設現在有個Borel measure μ, 我可以做一個measure μ_ε
μ_ε(E) =∫[ ∫ ρ_ε(x-y)dμ(y) ]dx
E
其中dx代表Lebesgue measure
可以簡單證明∫ ρ_ε(x-y)dμ(y) 對x是連續函數
現在假設有個φ是continuous with compact support,
則
(μ_ε, φ) = ∫[ ∫φ(x) ρ_ε(x-y)dμ(y) ]dx
= ∫[ ∫φ(x) ρ_ε(x-y)dx ]dμ(y) Fubini's theorem
= ∫[ ∫φ(x+y) ρ_ε(x)dx ]dμ(y) (translation invariance)
-> ∫φ(y) dμ(y) as ε-> 0, by uni. conv.
= (μ, φ)
因此μ_ε-> μ在weak-*下收斂, 隨著ε->0 (所以{μ_ε}稱為μ的regularization)
因為μ_ε是以一個連續函數的Lebesgue積分表示
而L^1 with strong topology又是seperable
所以能找到一個countable subset of continuous functions dense in L^1
既然dense in L^1, 用它Lebesgue積分得到的measure自然會dense在Borel measure中
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◆ From: 220.132.161.204
※ 編輯: willydp 來自: 220.132.161.204 (02/27 03:03)
※ 編輯: willydp 來自: 220.132.161.204 (02/27 03:03)