Re: [分析] 兩題請教

[kemowu (小展)]

※ 引述《kemowu (小展)》之銘言：
:
: Q2. Let f_n:E→R so that f_n→f pointwise on E, where E is an uncountable set.
: Prove that there is an infinite subset A of E so that f_n→f uniformly on A.
:
自己回一下看看對不對

Let ε>0 be given. Define

  E_n = {|f_n - f|≧ε} and A_N = ∪ E_n = {|f_n - f|≧ε for some n≧N}.
                                 n≧N

Note that A_N↘. Since f_n→f pointwise on E, we have
   ∞                    ∞
   ∩ A_N = empty, i.e., ∪ E-A_N = E.
   N=1                   N=1
Now since E is uncountable, there is an N such that E-A_N is infinite.

Put A = E-A_N. Then if n≧N, |f_n - f|<ε on A.
                                                      □

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推 ppia        :你這個方法剛好就是 Egoroff thm 的證明 一般來說
→ ppia        :for any e>0, f_n→f uniformly off E with m(E)<e,
→ ppia        :provided m(X)<infty, where X is the whole space.
→ ppia        :但是如果要證明這題 測度都不需用到 那個子集可數就
→ ppia        :好 可數就可以逐點操作再用對角線法..
→ ppia        :啊 我眼殘了 你上面的證明也沒有用到測度...