Re: [分析] characterization of weakly measurabl …
※ 引述《Rhymer (Design)》之銘言：
: Let (Ω,Σ) be a measurable space and X a Banach space.
: f:Ω->X is called weakly measurable if for each linear functional
: x' in X'(the norm dual of X),
: the scalar function < f, x' > is measurable.
: 類比於 storngly measurable 的定義, 我猜想
: f is weakly measurable iff there exists a sequence of step functions
: φ_n:Ω->X such that < φ_n, x' > converges to < f, x' > for each x' in X'.
: 不知道這個猜測是否正確?
: <= 方向的證明很簡單, 但是 => 卻沒有頭緒...
: 懇請板上神人指點迷津!
: 感謝!!
結果是否定的. 如果 φ_n -> f weakly a.e. 則 f 為 storngly measurable.
推論其實很簡單, 關鍵是 "Pettis' theorem" 以及
"subspace 的 weak closure 與 norm closure 是相同的" 這兩件事.
可惜我想不到...
以下是來自 Mathoverflow 的解答:
http://tinyurl.com/3jy9rzh
(Answered by Michael Renardy. Rephrased.)
If φ_n converges to f weakly a.e., then the range of f is contained in
a separable subspace of X (註). However, if f is weakly measurable, then by
Pettis' theorem, which states that f is strongly measurable iff it is
weakly measurable and its range is almost separable, f is strongly measurable.
(註) By Mazur's theorem, the closure and weak closure of a subspace in a
normed space is the same. See An introduction to Banach space theory
corollary 2.5.17 by Robert E. Megginson.
--
※ 發信站: 批踢踢實業坊(ptt.cc)
◆ From: 220.133.4.190
推
06/13 20:27, , 1F
06/13 20:27, 1F
→
06/14 08:17, , 2F
06/14 08:17, 2F
推
06/15 00:36, , 3F
06/15 00:36, 3F
→
06/15 00:37, , 4F
06/15 00:37, 4F
→
06/15 00:38, , 5F
06/15 00:38, 5F
→
06/15 08:19, , 6F
06/15 08:19, 6F