本文介绍了数学中的半代数、代数和σ-代数的概念。半代数满足包括全集在内的集合闭合、有限并集闭合的条件。代数则额外要求补集也属于代数。而σ-代数要求包含全集，有限交集和任意并集闭合。此外，σ-代数是代数，代数也是半代数。文章还探讨了这些结构的性质和生成方式，并给出了加性和σ-加性测度的例子。
Definitions
In this post, we define the semi-algebra, algebra, sigma-algebra.
Semi-algebra
Consider Ω \Omega Ω as the whole set (for example, Ω = R \Omega=\mathbb{R} Ω=R),
S ( Ω ) \mathcal{S}(\Omega) S(Ω) is the collection of subset of Ω \Omega Ω
DEF Semi-algebra S \mathscr{S} S is a subset of S ( Ω ) \mathcal{S}(\Omega) S(Ω) such that
- Ω ∈ S \Omega \in \mathscr{S} Ω∈S
- ∀ A , B ∈ S , A ∩ B ∈ S \forall A, B \in \mathscr{S}, A\cap B\in\mathscr{S} ∀A,B∈S,A∩B∈S
- ∀ A ∈ S , ∃ A 1 , A 2 , . . . , A n ∈ S s . t . A = ∑ i = 1 n A i \forall A\in\mathscr{S},\exists A_1, A_2,..., A_n\in\mathscr{S} s.t. A=\sum_{i=1}^nA_i ∀A∈S,∃A1,A2,...,An∈Ss.t.A=∑i=1nAi
[we define ∑ i A i \sum_i A_i ∑iAi in lecture 1]
Example:
{ ( a , b ] ⊆ R } \{(a, b]\subseteq \mathbb{R}\} { (a,b]⊆R}
(in fact, this example inspires the definition of semi-algebra)
Algebra
DEF Algebra A \mathscr{A} A is a subset of S ( Ω ) \mathcal{S}(\Omega) S(Ω) such that
- Ω ∈ A \Omega \in \mathscr{A} Ω∈A
- ∀ A , B ∈ A , A ∩ B ∈ A \forall A, B \in \mathscr{A}, A\cap B\in\mathscr{A} ∀A,B∈A,A∩B∈A
- ∀
最低0.47元/天 解锁文章
扫一扫
3434
被折叠的 条评论
为什么被折叠?
到【灌水乐园】发言
