[分析] 一題(outer measure)

[pop10353 (卡卡:目)]

Q:
   Show that if a set E has positive outer measure,then there is a bounded
subset of E that also has positive outer measure.

<pf>

Let In= [n,n+1] ,for each n in Ζ.
then E= ∪(E∩In).
By countable subadditivity that

0< m*(E)< Σ m*(E∩In)
        =

Thus,at least one of (E∩In) has positive outer measure.

It is bounded (subset of In) and is a subset
of E, so it is our desired set.

這是解法 ，應該是沒甚麼問題

但有疑問 如果這題我想要用 ~q -> ~p

那命題要怎麼改




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推 znmkhxrw    :假設不存在有界且正外測度的子集 => 有界的子集都是
→ znmkhxrw    :零測度 => 矛盾!
→ znmkhxrw    :(這邊子集都是指E的子集)
→ pop10353    :請問哪裡矛盾!?!?!
→ znmkhxrw    :0< m*(E)< Σ m*(E∩In) 阿
→ pop10353    :喔喔喔 可是ZNM大會錯意了 我是想要把p->q改成
→ pop10353    :~q->~p  而不是 p->~q