Re: [分析] 實變問題

[sato186 (銀色轟炸機)]

※ 引述《TrySoHard (夢永遠都只是夢)》之銘言：
: 1. Does there exist an infinite σ-algebra that has only countably
: many element?

　　Let G = {(a,b)∩Ｑ|a,b are both in Ｑ.}.

Then the σ-algebra generated by G is countable.

: 2. Let E be a measurable subset of [0,1]. Let A={ x^1/2  |  x is in E  }.
:    Is A a measurable subset of [0,1]?
: 先感謝各位解答了

　　Since taking square root is a continuous function on [0,1],

A is measurable.

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◆ From: 61.227.132.132
推 TrySoHard   :不懂1的那個例子為什麼是countable?
→ TrySoHard   :2.連續函數不保持measurable吧?
→ TrySoHard   :不是要Lipschitz才會保持measurable嗎?
推 yusd24      :你是對的
推 zombiea     :if Lipschitz preserve measurability... then
→ zombiea     :consider E_n=E ∩[1/n,1] then do it for each n
推 TrySoHard   :感謝樓上!! 那第一題呢? 還是看不懂QQ