[分析] 一個實變問題

[czk0622 (波波)]

A 是一個 R^n 的子集，不一定是 Lebesgue measurable

小弟想問，是否存在 A 使得 |A|_i < ∞, 但是|A|_e = ∞

|A|_i = sup|F| , F 是包含在 A 裡面的 closed set

|A|_e = inf|G| , G 是包住 A 的 open set

|.| 指的是 Lebesgue measure

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◆ From: 140.114.231.11
→ sato186     :Q^n
→ czk0622     :有理數是0測度耶...
→ Vulpix      :可是能包住有理點的最小開集就是全空間...
→ czk0622     :實數扣掉一些無理點也是open阿...
推 ppia        :＠＠ ...能包住有理點的開集體積應該可以任意小吧
→ ppia        :put r+A={r+a|a in A} 令Aㄈ[0,1]滿足
→ ppia        :(i)either r+A=s+A or r+A intersects s+A as empty
→ ppia        :for all rational s and r.
→ ppia        :(ii) [0,1]ㄈU(r+A), where the union taken all
→ ppia        :rational r between 0 and 1.
→ ppia        :上面這個集合在一般的教科書都有構造,會用到
→ ppia        :Axiom of Choice
→ ppia        :A的outer measure=M>0 否則[0,1]的outer measure =0
→ ppia        :注意 U(r+A)ㄈ[0,2] 所以
→ ppia        :A的inner measure:=m=0 否則[0,2]的inner measure=∞
→ ppia        :B:=U{n+A}, the union taken over all integral n
→ ppia        :就是你要的集合
→ ppia        :更正 "B:=U(n+A)"