推 chu1216 : 謝謝！！ 09/06 00:36
※ 引述《chu1216 (chu)》之銘言：
: 請問一下, 如下這道題該怎麼解?
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: 謝謝
Let W denote the Borel sigma-algrebra on the real numbers
Define T = { B in W : B is a subset of [0,1]}
M = { B∩[0,1]: B in W }
We'll prove the following two claims and combine them to show the conclusion.
Claim 1: M is a subset of T
Given X in M, there exists B in W such that X = B∩[0,1]
Obviously, X is a subset of [0,1] ........(1)
Since W is the sigma-algebra generated by the closed sets,
[0,1] belongs to W.
Because both B and [0,1] are in W, X = B∩[0,1] is in W .......(2)
(1),(2) → X is in T.
Claim 1 is proved
Claim 2: T is a subset of M
Given Y in T, Y is in W and Y is a subset of [0,1]. ......(3)
Since Y is a subset of [0,1], Y = Y∩[0,1]. ......(4)
(3),(4) → Y∩[0,1] is in M,
and thus Y = Y∩[0,1] is in M.
Claim 2 is proved
We have T = M by combining Claim 1 and 2
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※ 編輯: arrenwu (98.45.135.233 美國), 09/05/2021 17:01:40