[分析] 實變 Borel measurable

看板Math作者 (莉森)時間13年前 (2012/12/19 22:40), 編輯推噓4(4推 0噓 20→)
留言24則, 6人參與, 6年前最新討論串1/1
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 -- ※ 發信站: 批踢踢實業坊(ptt.cc) ◆ From: 140.113.25.147
推
12/20 00:27, , 1F
measurable 可以用 step function 逼近
12/20 00:27, 1F
→
12/20 00:27, , 2F
step function 可以用 continuous function 逼近
12/20 00:27, 2F
→
12/20 00:45, , 3F
不好意思 我這邊是Borel measurable function
12/20 00:45, 3F
→
12/20 01:02, , 4F
我了解你的意思了 答案是可行的
12/20 01:02, 4F
→
12/20 01:04, , 5F
但我想就目前有的知識推出結果
12/20 01:04, 5F
推
12/20 06:24, , 6F
把D定義成連續函數與其點收斂極限的集合
12/20 06:24, 6F
→
12/20 06:24, , 7F
證明D closed under poiontwise limits
12/20 06:24, 7F
→
12/20 06:26, , 8F
這跟證明當定義實數為有理數極限時，實數是完備的證
12/20 06:26, 8F
→
12/20 06:26, , 9F
名一樣
12/20 06:26, 9F
→
12/20 11:52, , 10F
thx^^
12/20 11:52, 10F
推
12/20 16:59, , 11F
converge a.s. 可以辦到
12/20 16:59, 11F
→
12/20 16:59, , 12F
但是 如果是要每個點就辦不到
12/20 16:59, 12F
→
12/20 17:00, , 13F
Royden實變的習題有提供一些資訊
12/20 17:00, 13F
→
12/20 17:01, , 14F
存在一Borel measurable function沒辦法用連續函數
12/20 17:01, 14F
→
12/20 17:01, , 15F
逐點逼近!
12/20 17:01, 15F
推
12/20 18:05, , 16F
yes... converge a.e. only
12/20 18:05, 16F
→
12/20 20:46, , 17F
thx 我去找Royden看看
12/20 20:46, 17F
→
12/20 20:52, , 18F
請問是在Royden的第幾頁? 我找不到
12/20 20:52, 18F
→
08/13 17:20, , 19F
逐點逼近! https://muxiv.com
08/13 17:20, 19F
→
09/17 15:14, , 20F
09/17 15:14, 20F
→
11/10 11:11, , 21F
//daxiv.com
11/10 11:11, 21F
→
11/10 11:11, , 22F
11/10 11:11, 22F
→
01/02 15:11, 7年前 , 23F
把D定義成連續函數與其 https://noxiv.com
01/02 15:11, 23F
→
07/07 10:24, 6年前 , 24F
證明D closed http://yaxiv.com
07/07 10:24, 24F
‣ 更多 lizzon 的文章
文章代碼(AID): #1GqT7Mqr (Math)