推 CNSaya :推 04/19 08:45
推 keroro321 :-) 04/19 09:15
推 herstein :詳細XD 04/19 09:40
※ 引述《ericakk (我還記得)》之銘言：
: 在看 Royden 的實變...在measurable space有提到complete..
: 然後看到以下的東西...這些事可以串起來的嗎??
:
: E = (－∞,∞), If f is monotone, then f is Borel measurable;
: hence, f is Lebesgure measurable.
:
: The completion of Borel measure is Lebesgure measure.
: 謝謝回答..
[我不知道我會不會造成你的困擾，看不懂就算了…]
(定義) A measure space (X, Σ, μ) is called complete if A ≦ B, where
B in Σ with μ(B) ＝ 0. Then A in Σ.
(定理) If (X, Σ, μ) is not complete, let
__
Σ ＝ { A ＝ B ∪ D: B in Σ, and D ≦ Z, where μ(Z)＝0 },
__
and define for such an A, let μ (A) ＝ μ (B).
Then
__ __ __
(i) Σ is a σ-algebra, and μ is a measure defined on Σ.
__ __
(ii) (X, Σ, μ) is a complete measure space.
(習題-等價) If (X, Σ, μ) is not complete, let
Σ ＝ { A ＝ B － D: B in Σ, and D ≦ Z, where μ(Z)＝0 },
__
and define for such an A, let μ (A) ＝ μ (B).
Then
__ __ __
(i) Σ is a σ-algebra, and μ is a measure defined on Σ.
__ __
(ii) (X, Σ, μ) is a complete measure space.
(習題) 給予外測度 μ^*, 則 (X,Σ,μ) 必為完備測度空間。
其中 Σ ＝ {μ^*-measurable sets}.
(習題) The completion of the Borel measure is Lebesgure measure.
NOTE. 這道習題得利用到 Lebesgue measurable set 的一些等價條件：
E in Ｌ ＜＝＞ E ＝ G_δ \ Z, where Z has Lebesgue measure zero.
Q:為什麼要給名稱 "完備 ?" 你可以想一想…
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