UA MATH523A 实分析2 测度论概念与定理整理

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UA MATH523A 实分析2 测度论概念与定理整理

UA MATH523A 实分析2 测度论概念与定理整理

• • σ \sigma σ-代数
  • 测度
  • 外测度
  • Borel测度

这一章主要的工作是在“性质良好”的集合上建立测度，思路是先讨论“性质良好”的集合的性质，也就是 σ \sigma σ-代数，然后公理化定义测度。解决问题需要具体的测度，构造一个测度并不是很容易，但我们可以通过构造外测度经过Caratheodory扩张得到测度。使用这个思路的一个经典的例子就是直线上的Borel测度。这就是这一章的主要内容，下面整理一下这一章已经建立起来的结果，需要注意的是这些是使得测度论比较完善的最少需要的概念与定理。

σ \sigma σ-代数

Algebra: closed under finite unions and complements.
σ \sigma σ-algebra: algebra that is closed under countable unions.
σ \sigma σ-algebra generated by E \mathcal{E} E: a unique smallest σ \sigma σ-algebra containing E \mathcal{E} E or say the intersections of all σ \sigma σ-algebras containing E \mathcal{E} E. Denoted as M ( E ) \mathcal{M}(\mathcal{E}) M(E).

Lemma 1.1 E ⊂ M ( F ) ⇒ M ( E ) ⊂ M ( F ) \mathcal{E} \subset \mathcal{M}(\mathcal{F}) \Rightarrow \mathcal{M}(\mathcal{E}) \subset \mathcal{M}(\mathcal{F}) E⊂M(F)⇒M(E)⊂M(F)

Borel σ \sigma σ-algebra: B ( X ) \mathcal{B}(X) B(X), where X X X is a metric space, generated by the family of all open sets in X X X.
G δ G_{\delta} Gδ​ set: countable intersection of open sets
F σ F_{\sigma} Fσ​ set: countable unions of closed sets
G δ σ G_{\delta \sigma} Gδσ​ set: countable union of G δ G_{\delta} Gδ​ set
F σ δ F_{\sigma \delta} Fσδ​ set: countable intersections of F σ F_{\sigma} Fσ​ set

Proposition 1.2 B ( R ) \mathcal{B}(\mathbb{R}) B(R) contains all open intervals, closed intervals, half-open intervals, open rays and closed rays.

Product σ \sigma σ algebra on product space X = ∏ α ∈ A X α X = \prod_{\alpha \in A}X_{\alpha} X=∏α∈A​Xα​, A A A countable index set
⊗ α ∈ A M α = { π α − 1 ( E α ) : E α ∈ M α , α ∈ A } \otimes_{\alpha \in A}\mathcal{M}_{\alpha}=\{\pi^{-1}_{\alpha}(E_{\alpha}):E_{\alpha} \in \mathcal{M}_{\alpha},\alpha \in A\} ⊗α∈A​Mα​={πα−1​(Eα​):Eα​∈Mα​,α∈A}where π α : X → X α \pi_{\alpha}:X \to X_{\alpha} πα​:X→Xα​ and M α \mathcal{M}_{\alpha} Mα​ sigma algebra on X α X_{\alpha} Xα​. Note that
π α − 1 ( E α ) = ∏ β ∈ A E β , E β = X β , ∀ β ≠ α \pi^{-1}_{\alpha}(E_{\alpha})=\prod_{\beta \in A} E_{\beta},E_{\beta}=X_{\beta},\forall \beta \ne \alpha πα−1​(Eα​)=β∈A∏​Eβ​,Eβ​=Xβ​,∀β​=α

Proposition 1.3 ⊗ α ∈ A M α = M ( E ^ ) , E ^ = { ∏ α ∈ A E α : E α ∈ M α } \otimes_{\alpha \in A}\mathcal{M}_{\alpha} = \mathcal{M}(\hat \mathcal{E}),\hat \mathcal{E} = \{\prod_{\alpha \in A}E_{\alpha}:E_{\alpha} \in \mathcal{M}_{\alpha}\} ⊗α∈A​Mα​=M(E^),E^={∏α∈A​Eα​:Eα​∈Mα​}

Proposition 1.4 ⊗ α ∈ A M ( E α ) = M ( F ^ 1 ) , F ^ 1 = { π α − 1 ( E α ) : E α ∈ M ( E α ) } \otimes_{\alpha \in A}\mathcal{M}(\mathcal{E}_{\alpha}) = \mathcal{M}(\hat \mathcal{F}_1),\hat \mathcal{F}_1 = \{\pi^{-1}_{\alpha}(E_{\alpha}):E_{\alpha} \in\mathcal{M}(\mathcal{E}_{\alpha})\} ⊗α∈A​M(Eα​)=M(F^1​),F^1​={πα−1​(Eα​):Eα​∈M(Eα