[分析] 實變 Borel measurable

[lizzon (莉森)]

小妹今天在一本書上看到Borel measurable的定義，如下

    ^
    C = the smallest collection of real-valued functions on R

        that contains the collection of coutinuous functions

        and is closed under pointwise limits

                   ^
    The members of C are called Borel measurable functions


問:
    Every Borel measurable function can approximated by

    continuous functions ?



我覺得是正確的 因為closed under pointwise limits

但書上找不到一樣或類似的性質

所以想問版上高手 看看是不是對的

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◆ From: 140.113.25.147
推 chy1010     :measurable 可以用 step function 逼近
→ chy1010     :step function 可以用 continuous function 逼近
→ lizzon      :不好意思  我這邊是Borel measurable function
→ lizzon      :我了解你的意思了 答案是可行的
→ lizzon      :但我想就目前有的知識推出結果
推 zombiea     :把D定義成連續函數與其點收斂極限的集合
→ zombiea     :證明D closed under poiontwise limits
→ zombiea     :這跟證明當定義實數為有理數極限時，實數是完備的證
→ zombiea     :名一樣
→ lizzon      :thx^^
推 cometic     :converge a.s. 可以辦到
→ cometic     :但是 如果是要每個點就辦不到
→ cometic     :Royden實變的習題有提供一些資訊
→ cometic     :存在一Borel measurable function沒辦法用連續函數
→ cometic     :逐點逼近!
推 chy1010     :yes... converge a.e. only
→ lizzon      :thx 我去找Royden看看
→ lizzon      :請問是在Royden的第幾頁? 我找不到
→ sneak       : 逐點逼近! https://muxiv.com
→ sneak       : 名一樣 https://daxiv.com
→ sneak       : //daxiv.com
→ sneak       :  https://daxiv.com
→ sneak       : 把D定義成連續函數與其 https://noxiv.com
→ muxiv       : 證明D closed  http://yaxiv.com