测度论与概率论基础

realhuhu 471 0

可测空间和可测映射

集合及其运算

集合

  1. 空间X:任意非空集合
  2. 集合A,B,C,\cdotsX子集空集\varnothing
  3. 元素xX的成员
  4. 属于\in:元素x属于A,记作x\in A;反之x \notin A
    • 指示函数
      I_A(x)= \begin{cases}1, & x \in A, \\0, & x \notin A\end{cases}
  5. A^cA^c\xlongequal{\text { def }}\{x: x \notin A\}
  6. 包含\subsetA\subset B\Leftrightarrow x \in A \Longrightarrow x \in B
  7. 等于=A=B\Leftrightarrow A\subset BB\subset A
  8. \cupA \cup B \xlongequal{\text { def }}\{x: x \in A \text {或} x \in B\}
  9. \capA \cap B \xlongequal{\text { def }}\{x: x \in A \text {且} x \in B\}
    • A\cap B=\varnothing,则A,B不交的
    • 并和交满足交换律结合律
      \begin{aligned} & (A \cup B) \cap C=(A \cap C) \cup(B \cap C) \\ & (A \cap B) \cup C=(A \cup C) \cap(B \cup C)\end{aligned}
  10. \backslashA \backslash B \xlongequal{\text { def }}\{x: x \in A \text {且} x \notin B\}
    • B\subset A,则A\backslash B真差
  11. 对称差 \DeltaA \Delta B \xlongequal{\text { def }}(A \backslash B) \cup(B \backslash A)

集族

  1. 集族\{A_t,t\in T\}T被称为指标集t被称为指标A_t为各不相同的集合
    • 若集族的集合两两间不交,则集族是两两不交
    • 集合序列\left\{A_n, n=1,2, \cdots\right\}
  2. \bigcup
    \bigcup_{t \in T} A_t \xlongequal{\text { def }}\left\{x: \exists t \in T\right.使\left.x \in A_t\right\}
  3. \bigcap
    \bigcap_{t \in T} A_t \xlongequal{\text { def }}\left\{x: x \in A_t, \forall t \in T\right\}

    • 交和并满足德摩根律
      \left\{\bigcup_{t \in T} A_t\right\}^c=\bigcap_{t \in T} A_t^c ; \quad\left\{\bigcap_{t \in T} A_t\right\}^c=\bigcup_{t \in T} A_t^c
  4. 非降A_n \uparrow
    A_n \subset A_{n+1}\text{总是成立}

    • 把集合\lim_{n \rightarrow \infty} A_n \xlongequal{\text { def }} \bigcup_{n=1}^{\infty} A_n称为它的极限
  5. 非增A_n \downarrow
    A_n \supset A_{n+1}\text{总是成立}

    • 把集合\lim_{n \rightarrow \infty} A_n \xlongequal{\text { def }} \bigcap_{n=1}^{\infty} A_n称为它的极限
    • 非降与非增的集合序列统称为单调序列,单调序列总有极限
  6. 上极限\liminf_{n \rightarrow \infty} A_n
    \liminf_{n \rightarrow \infty} A_n \xlongequal{\text { def }} \bigcup_{n=1}^{\infty} \bigcap_{k=n}^{\infty} A_k

    • 因为集合序列\{\bigcap_{k=n}^{\infty} A_k,n=1,2,\cdots\}是非降的
    • x\in\liminf_{n \rightarrow \infty} A_n表示:对于充分大的n\forall k\ge n,x\in \bigcap_{k=n}^{\infty} A_k成立
  7. 下极限\limsup_{n \rightarrow \infty} A_n
    \limsup_{n \rightarrow \infty} A_n \xlongequal{\text { def }} \bigcap_{n=1}^{\infty} \bigcup_{k=n}^{\infty} A_k

    • 因为集合序列\{ \bigcup_{k=n}^{\infty} A_k,n=1,2,\cdots\}是非增的
    • x\in\limsup_{n \rightarrow \infty} A_n表示:对于充分大的n\exists k\ge n,x\in\bigcup_{k=n}^{\infty} A_k成立
  8. 一般情况下的极限\lim_{n \rightarrow \infty} A_n
    \lim_{n \rightarrow \infty} A_n \xlongequal{\text { def }} \liminf_{n \rightarrow \infty} A_n=\lim_{n \rightarrow \infty} \sup A_n

    • 上极限是下极限的子集:\liminf_{n \rightarrow \infty} A_n \subset \limsup_{n \rightarrow \infty} A_n
    • 当上极限等于下极限则集合序列极限存在

集合系

  1. 集合系\mathscr{A},\mathscr{B},\cdots:由X的集合构成的集合
  2. \pi\mathscr{P}:对交运算封闭的集合系
    A, B \in \mathscr{P} \Longrightarrow A \cap B \in \mathscr{P}

    • 实数空间\boldsymbol{R}上的\pi系:\mathscr{P}_{\boldsymbol{R}} \xlongequal{\text { def }}\{(-\infty, a]: a \in \boldsymbol{R}\}
  3. 半环\mathscr{Q}:集合的真差可以由两两不相交集合的可列并表示的π
    \forall A,B\in\mathscr{Q},\ A \supset B,\exists \left\{C_k \in \mathscr{Q},\ k=1, \cdots, n\right\}, C_i\cap C_j=\varnothing\quad \mathrm{s.t.}\quad A \backslash B=\bigcup_{k=1}^n C_k

    • 实数空间\boldsymbol{R}上的半环:\mathscr{Q}_{\boldsymbol{R}} \xlongequal{\text { def }}\{(a, b]: a, b \in \boldsymbol{R}\}
  4. \mathscr{R}:对并和差运算封闭的半环
    A, B \in \mathscr{R} \Longrightarrow A \cup B, A \backslash B \in \mathscr{R}

    • 实数空间\boldsymbol{R}上的环:\mathscr{R}_{\boldsymbol{R}} \xlongequal{\text { def }} \bigcup_{n=1}^{\infty}\left\{\bigcup_{k=1}^n\left(a_k, b_k\right]: a_k, b_k \in \boldsymbol{R}\right\}
    • \sigma:对无限并和差运算封闭的环
      \begin{aligned} & A, B \in \mathscr{R} \Longrightarrow A \backslash B \in \mathscr{R} ; \\ & A_n \in \mathscr{R}, n=1,2, \cdots \Longrightarrow \bigcup_{n=1}^{\infty} A_n \in \mathscr{R} .\end{aligned}
  5. \mathscr{A}:包含X、对余运算封闭的环
    X \in \mathscr{A} ; A \in \mathscr{A} \Longrightarrow A^c \in \mathscr{A}

    • 也叫做代数
  6. 单调系\mathscr{M}:集合系中任意集合构成的单调序列的极限也属于集合系
    A_n\in \mathscr{M}\text{且}\{A_n,n=1,2,\cdots\}\text{单调}\Longrightarrow \lim A_n \in \mathscr{M}
  7. \lambda\mathscr{L}:满足下列条件的单调系
    \begin{aligned} & X \in \mathscr{L} ; \\ & A, B \in \mathscr{L} \text {且} A \supset B \Longrightarrow A \backslash B \in \mathscr{L} ; \\ & A_n \in \mathscr{L} \text {且} A_n \uparrow \Longrightarrow \bigcup^{\infty} A_n \in \mathscr{L} .\end{aligned}
  8. \sigma\mathscr{F}:满足下列条件的\lambda
    \begin{aligned} & X \in \mathscr{F} ; \\ & A \in \mathscr{F} \Longrightarrow A^c \in \mathscr{F} ; \\ & A_n \in \mathscr{F}, n=1,2, \cdots \Longrightarrow \bigcup_{n=1}^{\infty} A_n \in \mathscr{F} .\end{aligned}

    • 也叫做\sigma代数
    • 最小的\sigma域:\{\varnothing, X\};最大的\sigma域:\{A: A \subset X\}
    • 可测空间:(X, \mathscr{F})
    • 既是单调系又是域的集合系必是\sigma域:\mathscr{M}+\mathscr{A}=\mathscr{F}
    • 既是\lambda系又是\pi系系的集合系必是\sigma域:\mathscr{L}+\mathscr{P}=\mathscr{F}
    • 包含X\sigma环是\sigma

σ域的生成

  1. 生成(环、单调系、\lambda系、\sigma域)\mathscr{S}:若集合系\mathscr{E}\subset\mathscr{S},且任意包含\mathscr{E}的(环、单调系、\lambda系、\sigma域)也包含\mathscr{S},则\mathscr{S}\mathscr{E}生成
    • \mathscr{S}是包含\mathscr{E}的最小的(环、单调系、\lambda系、\sigma域)
  2. 任意\mathscr{E}都存在生成的环、单调系、\lambda系、\sigma域,记作r(\mathscr{E}),m(\mathscr{E}),l(\mathscr{E}),\sigma(\mathscr{E})
    • 半环\mathscr{Q}的生成环为
      r(\mathscr{Q})=\bigcup_{n=1}^{\infty}\left\{\bigcup_{k=1}^n A_k:\left\{A_k \in \mathscr{Q}, k=1, \cdots, n\right\}\text{两两不交}\right\}
    • \mathscr{A}的生成单调系就是生成\sigma
      \sigma(\mathscr{A})=m(\mathscr{A})
    • \pi\mathscr{P}的生成\lambda系就是生成\sigma
      \sigma(\mathscr{P})=l(\mathscr{P})
    • \mathscr{A}包含于单调系\mathscr{M},则\mathscr{A}的生成\sigma域也包含于\mathscr{M}\pi\mathscr{P}包含于\lambda\mathscr{L},则\mathscr{P}的生成\sigma域也包含于\mathscr{L}
      \begin{align} &\mathscr{A} \subset \mathscr{M} \Longrightarrow \sigma(\mathscr{A}) \subset \mathscr{M} \\ &\mathscr{P} \subset \mathscr{L} \Longrightarrow \sigma(\mathscr{P}) \subset \mathscr{L} \end{align}
  3. \mathbf{R}上的Borel集合系\mathscr{B}_R
    \mathscr{B}_{\boldsymbol{R}} \xlongequal{\text { def }} \sigma\left(\mathscr{Q}_{\boldsymbol{R}}\right)=\sigma\left(\mathscr{P}_{\boldsymbol{R}}\right)

    • \mathscr{B}_R中的集合称为\mathbf{R}上的Borel集
    • \mathbf{R}中开集组成的集合系\mathscr{O}_R满足
      \mathscr{B}_R=\sigma\left(\mathscr{O}_R\right)
  4. 拓扑可测空间(X, \mathscr{B})
    • 对于拓扑空间X,以\mathscr{O}记其开集系,则X上的Borel集合系为
      \mathscr{B} \xlongequal{\text { def }} \sigma(\mathscr{O})

可测映射和可测函数

  1. 映射f:对于集合X,Y,对于每个x\in X,存在唯一f(x)\in Y
    • 又叫做XY的映射定义在X上取值于Y的函数
    • f(x)称为fx
  2. 原像f^{-1} B X中映射后属于B的子集
    f^{-1} B \xlongequal{\text { def }}\{f \in B\}=\{x: {f(x)} \in B\}\quad\quad(B\subset Y)

    • 性质
      \begin{aligned} & f^{-1} \varnothing=\varnothing ; \quad f^{-1} Y=X ; \\ & B_1\subset B_2\Longrightarrow f^{-1} B_1\subset f^{-1} B_2; \\ & \left(f^{-1} B\right)^c=f^{-1} B^c, \quad \forall B \subset Y; \\ &f^{-1} \bigcup_{t \in T} A_t=\bigcup_{t \in T} f^{-1} A_t, \quad \forall\left\{A_t \subset Y, t \in T\right\}; \\ &f^{-1} \bigcap_{t \in T} A_t=\bigcap_{t \in T} f^{-1} A_t, \quad \forall\left\{A_t \subset Y, t \in T\right\}; \end{aligned}\\
  3. \sigma\left(f^{-1} \mathscr{E}\right)=f^{-1} \sigma(\mathscr{E})
    • 对于可测空间(X, \mathscr{F})(Y, \mathscr{S})以及XF的映射f,如果
      f^{-1} \mathscr{S} \subset \mathscr{F}
      就把f(X, \mathscr{F})(Y, \mathscr{S})可测映射随机元
      \sigma(f) \xlongequal{\text { def }}f^{-1} \mathscr{S} 叫做使f可测的最小\sigma
    • 性质
      f(X, \mathscr{F})(Y, \sigma(\mathscr{E}))的可测映射\Leftrightarrow f^{-1} \mathscr{E} \subset \mathscr{F}
      f,g(X, \mathscr{F})(Y, \mathscr{S})(Y, \mathscr{S})(Z, \mathscr{Z})的可测映射,则f \circ g(X, \mathscr{F})(Z, \mathscr{Z})的可测映射
  4. 广义实数集\overline{\boldsymbol{R}}:将无穷看做数,可以作为端点,可以参与运算
    \overline{\boldsymbol{R}} \xlongequal{\text { def }} \boldsymbol{R} \cup\{-\infty\} \cup\{\infty\}

    • a=a^{+}-a^{-}
  5. 广义Borel集合系\mathscr{B}_{\bar{\boldsymbol R}}
    \mathscr{B}_{\bar{\boldsymbol R}} \xlongequal{\text { def }} \sigma\left(\mathscr{B}_{\boldsymbol{R}},\{-\infty\},\{\infty\}\right)

    • 下列命题成立
      \begin{aligned} \mathscr{B}_{\bar{\boldsymbol R}} & =\sigma([-\infty, a): a \in \boldsymbol{R}) \\ & =\sigma([-\infty, a]: a \in \boldsymbol{R}) \\ & =\sigma((a, \infty]: a \in \boldsymbol{R}) \\ & =\sigma([a, \infty]: a \in \boldsymbol{R}) .\end{aligned}
    • (X, \mathscr{F})(\bar{\boldsymbol R}, \mathscr{B}_{\bar{\boldsymbol{R}}})f称为(X, \mathscr{F})上的可测函数
    • (X, \mathscr{F})({\boldsymbol R}, \mathscr{B}_{{\boldsymbol{R}}})f称为(X, \mathscr{F})上的有限值可测函数随机变量
    • 下列说法等价
      f(X, \mathscr{F})上的可测函数(或随机变量)
      \{f\lt a\} \in \mathscr{F}, \forall a \in \boldsymbol{R}
      \{f \leqslant a\} \in \mathscr{F}, \forall a \in \boldsymbol{R}
      \{f\gt a\} \in \mathscr{F}, \forall a \in \boldsymbol{R}
      \{f \geqslant a\} \in \mathscr{F}, \forall a \in \boldsymbol{R}
      \{f=a\} \in \mathscr{F}, \forall a \in \overline{\boldsymbol{R}}
    • f,g是可测函数,则\{f\lt g\},\{f \leqslant g\},\{f=g\} \in \mathscr{F}

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