推 sato186 :Nice Play!! 我只想到E_m,n那邊 07/28 04:28
※ 引述《kemowu (小展)》之銘言：
: Q1. Let E_n be Lebesgue measurable subsets of [0,1] and m(E_n)→1, where m is
: the Lebesgue measure. Show that there is a subsequence {E_(n_j)} so that
: ∞
: m( ∩ E ) > 0.
: j=1 n_j
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.
set f'n:= f-fn, so we may assume f=0
set En,m = { x | |fm'(x) |<1/n for all m' larger than m}
note that for fixed n, for any x in E, then there exists m_n(x) in N such that
x in En,m_n(x) by pointwise convergence of fn on E.
Since E is uncountable, hence there exists m1 in N such that
there are uncountable many x in E with m_1(x) =m1.
Set E_1 ={x in E| m_1(x) =m1 }
Then define E_2 :={ x in E_1 | m_2(x) =m2 } for some m2 in N
and E_2 is uncountable. Defining E_i by induction on this manner.
So we have E_1 containing E_2 containing E_3 ....
For each E_i is uncountable , hence we have pick a_i from each E_i such that
{a_i} are all distinct.
Claim: A:={a_i} is what we need.
Note that checking epsilon =1/n for any n is sufficient.
for 1/n given, then for a_k, k≧n, m larger than m_n, then
|fm(a_k)| < 1/n . Then pick m' larger than m_n
such that a_1,.. a_(k-1) satisfy |fm(a_i)|<1/n for i=1,..,k-1 for all
m larger than m'.
Q.E.D.
Supplement
1. I have no idea whether A can be chosen to be uncountable or not.
2. Uncountable assumption is necessary, consider E=N and fn(x) =x/n
PS thanks Lim to point out a_i to me.
--
※ 發信站: 批踢踢實業坊(ptt.cc)
◆ From: 114.43.100.130
※ 編輯: zombiea 來自: 114.43.100.130 (07/28 03:51)