高等概率论

集合论

集合运算

基本运算

\(\text{有限交} \iff \bigcap_{i=1}^n A_i\)

\(\text{可列交} \iff \bigcap_{i=1}^{\infty} A_i\)

\(\text{不交并} \iff \biguplus_{t\in T} A_t\)

\(\text{有限并} \iff \bigcup_{i=1}^n A_i\)

\(\text{可列并} \iff \bigcup_{i=1}^{\infty} A_i\)

\(\text{补集 (complement)} \iff A^c:=\Omega\backslash A\)

\(\text{差集} \iff A \backslash B = A\cap B^c \iff \{\omega:\omega\in A, \omega\notin B\}\)

\(\text{对称差 (symmetric diference)} \iff A\triangle B = (A\backslash B)\cup(B\backslash A) = (A\cup B)\backslash(A\cap B)\)

\(\text{分配率 (distributive law)} \iff A \cap \left( \bigcup_{t \in T} B_t \right) = \bigcup_{t \in T} (A \cap B_t)\)

\(\text{分配率 (distributive law)} \iff A \cup \left( \bigcap_{t \in T} B_t \right) = \bigcap_{t \in T} (A \cup B_t);\)

\(\text{de Morgan's laws} \iff \left( \bigcup_{t \in T} A_t \right)^c = \bigcap_{t \in T} A_t^c\)

\(\text{de Morgan's laws} \iff \left( \bigcap_{t \in T} A_t \right)^c = \bigcup_{t \in T} A_t^c\)

上极限和下极限

上极限

\[\begin{aligned} \omega\in\limsup_{n\to\infty} A_n &\iff\omega\in\bigcap_{n=1}^{\infty}\bigcup_{k=n}^{\infty} A_k\\ &\iff\forall n\geq 1,\omega\in\bigcup_{k=n}^{\infty} A_k\\ &\iff\forall n\geq 1,\exist k \geq n,\text{ s.t. }\omega\in A_k \\ &\iff\omega\in\{A_n,\text{ i.o. }\} \end{aligned} \]

下极限

\[\begin{aligned} \omega\in\liminf_{n\to\infty} A_n &\iff\omega\in\bigcup_{n=1}^{\infty}\bigcap_{k=n}^{\infty} A_k\\ &\iff\exist n\geq 1,\omega\in\bigcup_{k=n}^{\infty} A_k\\ &\iff\exist n\geq 1,\forall k \geq n,\text{ s.t. }\omega\in A_k \\ &\iff\omega\in\{A_n,\text{ ult. }\} \end{aligned} \]

下极限含于上极限

\[\liminf_{n\to\infty} A_n \subset \limsup_{n\to\infty} A_n \]

集合的极限

\[\begin{aligned} A_n \text{收敛于} A &\iff \liminf_{n\to\infty} A_n = \limsup_{n\to\infty} A_n = A\\ &\iff \lim_{n\to\infty} A_n = A\\ &\iff A_n \to A\quad (n\to\infty) \end{aligned} \]

单调集

\[A_n\uparrow \iff A_1\subset A_2\subset\cdots\subset A_n\subset\cdots \implies \lim_{n\to\infty}A_n=\bigcup_{n=1}^{\infty} A_n \]

\[A_n\downarrow \iff A_1\supset A_2\supset\cdots\supset A_n\supset\cdots \implies \lim_{n\to\infty}A_n=\bigcap_{n=1}^{\infty} A_n \]

集类

\( \text{集类 (class of sets)} \iff \text{集族 (collection of sets)} \iff \text{集合的集合} \)

\( \text{集类中的元素} \iff \text{集类中的集合} \)

\( \mathcal{A}_1 \text{ 比 } \mathcal{A}_2 \text{ 小 } \iff \mathcal{A}_1 \subset \mathcal{A}_2 \iff \mathcal{A}_1 \text{ 是 } \mathcal{A}_2 \text{ 的子集类 } \)

\( \mathcal{A}_1 \text{ 比 } \mathcal{A}_2 \text{ 大 } \iff \mathcal{A}_1 \supset \mathcal{A}_2 \iff \mathcal{A}_2 \text{ 是 } \mathcal{A}_1 \text{ 的子集类 } \)

\[\text{一族集类 } \{\mathcal{A}_t,t\in T\} \text{ 的交} \iff \bigcap_{t\in T} A_t = \{A:A\in\mathcal{A}_t, \forall t\in T\} \]

\[\text{一族集类 } \{\mathcal{A}_t,t\in T\} \text{ 的并} \iff \bigcup_{t\in T} A_t = \{A:A\in\mathcal{A}_t, \exist t\in T\} \]

\( \text{幂集 (power set)} \iff \mathcal{P}(\Omega) \iff \Omega \text{ 的所有子集} \iff \Omega \text{ 上的最大集类} \)

\[\#\mathcal{P}(\Omega) = 2^{\#\Omega} \iff \mathcal{P}(\Omega) \text{ 中的元素的个数} \]

映射、笛卡尔积与逆像

映射

\( \text{映射 (mapping)} \iff f:X\to Y \iff \forall x\in X,\exist y\in Y, \text{ s.t. } y=f(x) \)

\( \text{变换 (transformation)} \iff f:X\to X \)

\( \text{像 (image)} \iff \)

\( \text{原像 (preimage)} \iff \text{逆象 (inverse image)} \iff \)

\( \text{定义域 (domain if definition)} \iff \)

\( \text{值域 (range)} \iff \)

\( \text{值空间 (range space)} \iff \)

\( \text{(实值)函数} \iff \)

\( \text{复值函数} \iff \)

\( \text{单射 (injective, one-to-one)} \iff \)

\( \text{满射 (surjective, onto)} \iff \)

\( \text{双射 (bijective)} \iff \)

\( \text{逆映射 (inverse mapping)} \iff \)

\( \text{复合 (composition)} \iff \)

示性函数

\[\text{示性函数 (indicator function)} \iff I_A(\omega)= \begin{cases} 1,&\omega\in A\\ 0,&\omega\notin A \end{cases} \]

\[I_A=I_B \iff A = B \]

\( \text{上端} \iff a\vee b = \max\{a,b\} \)

\( \text{下端} \iff a\wedge b = \min\{a,b\} \)

笛卡尔积

\( \text{笛卡尔积 (Cartesian product)} \iff \)

\( \text{欧式空间} \iff \)

\( \text{欧式平面} \iff \)

\( \text{实数空间} \iff \)

\( \text{有限矩形} \iff \)

\( \text{无限矩形} \iff \)

\( \text{乘积空间 (product space)} \iff \)

\( \text{坐标映射 (coordinate mapping)} \iff \)

\( \text{投影映射 (projective mapping)} \iff \)

逆像

\( \text{逆像} \iff \)

\( \text{相容的 (compatible)} \iff \)

\( \text{坐标映射} \iff \)

\( \text{坐标函数} \iff \)

\( \text{均值定理} \iff \)

集合的势

Bernstenin 定理

\( \text{对等 (equipollence)} \iff A \sim B \)

\( \text{势 (cardinality)} \iff \text{基数 (cardinal number)} \iff \)

\( \text{有限集 (finite set)} \iff \)

\( \text{无限集 (infinite set)} \iff \)

\( \text{ ()} \iff A \prec B \)

\( \text{ ()} \iff A \succ B \)

\( \text{Bernstenin 定理} \iff \)

可数集与不可数集

\( \text{可数集 (countable infinite set)} \iff \)

\( \text{至多可数集 (at most countable set)} \iff \)

\( \text{不可数集 (uncountable set)} \iff \)

\( \text{Cantor 连续统假设 (cantinuum hypothesis)} \iff \)

\( \text{Cantor 定理} \iff \)

\( \text{划分 (partition)} \iff \)

点集拓扑学

度量空间

\( \text{度量} \iff \)

\( \text{度量空间} \iff \text{距离空间} \iff \)

\( \text{离散度量} \iff \)

\( \text{有界函数空间} \iff \)

\( \text{连续函数空间} \iff C[a,b] \iff \)

\( \text{实数空间} \iff \mathbb{R} \iff \)

开集与领域

\( \text{球形领域 (spherical neighborhood)} \iff \)

\( \text{开集 (open set)} \iff \rho \text{ -开集} \)

\( \text{领域} \iff \)

完备度量空间

\( \text{基本列 (fundamental sequence)} \iff \)

\( \text{完备的 (complete)} \iff \)

Banach 空间

\( \text{实线性空间 (real linear space)} \iff \)

\( \text{范数 (norm)} \iff \)

\( \text{赋范线性空间 (normed linear space)} \iff \)

\( \text{完备赋范空间 (Bannach space)} \iff \)

乘积赋范空间

\( \text{乘积赋范空间} \iff \)

\( \text{有限维欧氏空间的完备性} \iff \)

\( \text{复空间的完备性} \iff \)

\( \text{可数无穷维欧氏空间的完备性} \iff \)

\( \text{Cauchy-Schwarz 不等式} \iff \)

\( \text{马氏距离 (Mahalanobis distance)} \iff \)

拓扑空间

拓扑空间

\( \text{拓扑 (topology)} \iff \)

\( \text{拓扑空间} \iff \)

\( \text{开集 (open set)} \iff \)

\( \text{闭集 (closed set)} \iff \)

\( \mathcal{T}_1 \text{ 粗于 } \mathcal{T}_2 \iff \)

\( \mathcal{T}_1 \text{ 细于 } \mathcal{T}_2 \iff \)

\( \text{平凡拓扑} \iff \)

\( \text{离散拓扑 (discrete topology)} \iff \)

\( \text{度量拓扑} \iff \)

\( \text{通常拓扑 (usual topology)} \iff \)

领域

\( \text{领域} \iff \)

\( \text{开领域 (open neighborhood)} \iff \)

\( \text{领域系 (neighborhood system)} \iff \)

基

\( \text{基 (base)} \iff \)

\( \text{以 } \mathcal{B} \text{ 为基生成的拓扑} \iff \)

\( \text{可数基} \iff \)

子基

\( \text{子基 (sub base)} \iff \)

\( \text{以 } \mathcal{S} \text{ 为子基生成的拓扑} \iff \)

制作新拓扑空间的方法

\( \text{子空间} \iff \)

\( \text{诱导拓扑 (induced topology)} \iff \)

\( \text{诱导开集} \iff \)

\( \text{Tychonoff 乘积拓扑} \iff \)

\( \text{Tychonoff 乘积空间} \iff \)

\( \text{坐标空间} \iff \)

\( \text{标准基} \iff \)

\( \text{聚点 (accumulation point)} \iff \)

\( \text{导集 (derived set)} \iff \)

\( \text{内点 (interior)} \iff \)

\( \text{边界点 (boundary point)} \iff \)

\( \text{边界 (boundary)} \iff \)

\( \text{附着点} \iff \)

连续映射

度量空间上的连续映射

\( \rho - \sigma\text{ 连续的 (continuous)} \iff \)

\( \text{上确界范数} \iff \)

\( \text{开映射} \iff \)

\( \text{闭映射} \iff \)

\( \text{同胚映射} \iff \)

\( \text{同胚 (homeomorphism)} \iff \)

\( \text{保距的 (distance preserving)} \iff \)

拓扑空间上的连续映射

可数性和可分性

分离性

紧性

度量空间中的紧性特征

集类

常见集类

半环

代数

\(\sigma\) 代数

单调类

\(\lambda\) 类

\( \text{ ()} \iff \)

\( \text{ ()} \iff \)

\( \text{ ()} \iff \)

\( \text{ ()} \iff \)

\( \text{ ()} \iff \)

\[\begin{aligned} &X\text{ 是 r.v.}\\ \iff& X\in\mathcal{L}(\Omega,\mathcal{F})\\ \iff& X:(\Omega,\mathcal{F})\to(\mathbb{R},\mathcal{B}(\mathbb{R})) \text{ 是可测映射}\\ \iff& \sigma(X)=X^{-1}(\mathcal{B}(\mathbb{R}))\subset\mathcal{F}\\ \iff& \forall x\in\mathbb{R}, X^{-1}(-\infty,x]\in\mathcal{F}\\ \iff& \forall x\in\mathbb{R},\{\omega\in\Omega : X(\omega)\leq x\}\in\mathcal{F}\\ \iff& \forall x\in\mathbb{R},\{X\leq x\}\in\mathcal{F} \end{aligned} \]

\( \text{ ()} \iff \)

\( \text{ ()} \iff \)