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作者  b218h (Gordon (In Canada)) 站內  ALGEBRA
標題  [問題] Royden Real Analysis page 19. no.20
時間  2008/06/21 Sat 15:06:33

Real Analysis by Royden, Third Edition.

Section 1.4 page 19 no. 20

Problem : Let X be a set.

          Let K be a collection of subsets of X and E be an element in the

          σ-algebra generated by K. Then there is a countable subcollection

          K_0 contained in K s.t. E is an element of the σ-algebra A_0

          generated by K_0.


[pf]
          *
    Let K = {countable subsets of K}

                ╭        │             * ╮
        H = ∪┤σ(K_0)│K_0 is in K  ├
                ╰        │               ╯

Claim : H is a sigma-algebra of subsets of X

(i) ψ is in Η. o.k
                                                                   *
(ii) Let Α be in Η, then Α is in σ(K_0) for some K_0 is in K

                                   c
     =>                           Α is in σ(K_0)
                                   c
     =>                           Α is in Η

(iii) Let {Α_i} be a sequence of sets in Η

      Then, for each i, there is a corresponding index λ in Λ where Λ is

countable s.t.                                    ┌             ┐
                        Α_i is in σ(K_λ) C σ│   ∪   K_λ│
                                              ̄   └λ in Λ     ┘

                                                               *
Since                       ∪   K_λ is countable, it's in K
                         λ in Λ


                        ∞              ┌             ┐
=>                      ∪ Α_i is in σ│   ∪   K_λ│C Η
                       i=1             └λ in Λ     ┘ ̄


H is a sigma-algebra as shown.


Claim : σ(K) C H

Now, we first show that K C H
                                       *
    (pf) Let X be in K, then {X} is in K

         X is in σ({Χ}) C H
                            ̄
Recall the definition of a sigma-algebra.

              def   ┌   │                                          ┐
       σ(K) === ∩│ F│F contains K, and F is a sigma-algebra│
                    └   │                                          ┘

              C H


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發信站 [中央數學  織夢天堂 bbs.math.ncu.edu.tw]
  •FROM [b218h 從 d154-20-178-83.bchsia.telus.net 發表]
□ Modify: 2008/06/21 Sat 15:12:42  d154-20-178-83.bchsia.telus.net 修改
→ VaTin :懶的看XD 能把絕的不確定的句子點出來嗎?                   08/06/21
→ b218h :歹勢忘了講 sigma(K)為 sigma-algebra gen. by K            08/06/21
→ hlgau :(i)錯掉了,右邊只是聯集,沒有再生成,是不會等於左邊的!      08/06/21
→ b218h :我懂老師的意思了 :)
→ b218h :可以再幫我看看 修正得對不對嗎^^?                         08/06/22
→ hlgau :OK!  很好!!                                              08/06/22
→ b218h :yeah. 感謝高大指教 :)                                    08/06/22
□ Modify: 2008/06/22 Sun 11:29:07  d154-20-178-83.bchsia.telus.net 修改

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