- Chapter 1 Measure spaces
- Chapter 2 Integration
- Chapter 3 Spaces of integrable functions
- Chapter 4 Hilbert spaces
- Chapter 5 Fourier series
- Chapter 6 Operations on measures
- Chapter 7 The fundamental theorem of the integral calculus
- Chapter 8 Measurable transformations
- Chapter 9 General concepts of Probability
- Chapter 10 Conditional probability and independece
- Chapter 11 Convergence of random variables
- Chapter 12 Sequences of independent variables
- Chapter 13 Stationary sequences and elements of ergodic theory
Luigi Ambrosio, Giuseppe Da Prato, Andrea Mennucci, An Introduction to Measure Theory and Probability.
Chapter 1 Measure spaces
Index:
- ring/algebras P2
- \(\sigma\)-algebras P3
- Borel \(\sigma\)-algebras P3
- \(\sigma\)-additive P4
- \((X,\mathscr{E},\mu)\) P7
- finite, \(\sigma\)-finite P7
- \(\mathscr{E}_{\mu}\), \(\mu-\)completion P8
- \(\pi-\)systems P9
- Dynkin-systems P10
- Outer measure P11
- \(\mathscr{S}:=\{(a,b]:a<b \in \mathbb{R}\}\) P12
- Lebesgue measure \(\lambda\) P12
P9页的Caratheodory定理是在环\(\mathscr{E}\)的基础上建立的(实际上半环足以), 通过半环生成\(\sigma\)域(通过\(\sigma(\mathscr{K})=\mathscr{D}(\mathscr{K})\)). 通过\(\mathscr{E}\)构建可测集域(外测度, 扩张), 由于\(\sigma(\mathscr{E})\)也是可测集, 所以满足所需的可加性. 当定义在\(\mathscr{E}\)的测度\(\mu\)是\(\sigma\)有限的时候(或者存在一个分割), 这个扩张是唯一的.
Chapter 2 Integration
Index:
- Inverse image \(\varphi^{-1}(I)\) P23
- \((\mathscr{E}, \mathscr{F})\)-measureable P23
- canonical representation of \(\varphi\) P25
\]
- repartition function P28
- archimedean integral P30
- \(\mu\)-integrable P32
- \(\mu\)-uniformly integrable P37
什么是可测函数, 以及什么是\(\mathscr{E}\)-可测函数是很重要的 (P24).
什么是\(\mu\)-integrable也是很重要的(在\(\mathscr{E}\)-可测函数定义的).
不同于我看到的一般的积分的定义, 这一节是从 repartition function 和 archimedean integral入手的, 特别是
\]
的定义式非常之有趣.
Chapter 3 Spaces of integrable functions
Index:
- \(L^p\),\(\mathcal{L}^p\) P44
- equivalence class \(\tilde{\varphi}\)
- Legendre transform P45
- \(\mu\)-essentially bounded P45
- Jensen inequality P45
- \(C_b\) P54
首先需要注意的是, \(L^p\)空间是定义在\(\mu\)-integrable上的, 所以其针对值域为\((\mathbb{R},\mathscr{B}(\mathbb{R}))\).
Chapter 4 Hilbert spaces
Index:
- Orthonormal system P63
- Complete orthonormal system P64
- Separable P64
- pre-Hilbert space P57
- Hilbert space (complete) P58
投影定理, 子空间或者凸闭集(条件和结论需要调整).
Chapter 5 Fourier series
Index:
- "Heaviside" function P71
- totally convergent P75
Chapter 6 Operations on measures
Index:
- Measureable rectangle P79
- sections, \(E_x,E^y\) P79
- dimensional constant \(w_n=\mathcal{L}^n(B(0,1))\) p83
- \(\delta\)-box P84
- cylindrical set P86
- concentrated set P92
- singular measures P92
- total variation P97
- stieltjes integral P103
- weak convergence P103
- Tightness of measures P104
- Fourier transform P108
这一章很重要!
Part1: Fubini-Tonelli
Part2: Lebesgue分解定理P92
Part3: Signed measures
Part4: \(F(x):= \mu((-\infty,x])\), P102, 弱收敛 \(\lim_{h\rightarrow \infty}\mu_h(-\infty, x]=\mu((-\infty, x])\) (除去可数多个点)
Part5: Fourier transform, 以及测度的Fourier transform (后面概率的表示函数有用), Levy定理P112.
Chapter 7 The fundamental theorem of the integral calculus
Index:
- density points, rarefaction points P121
- Heaviside function P121
- Cantor-Vitali function P121
- total variation P116
\]
\]
Chapter 8 Measurable transformations
Index:
- differential P123
- Jacobian determinant P125
- diffeomorphism P125
- critical set \(C_F\) P125
\]
有一个问题就是，我看其理论都是限制在非负函数上的, 但是个人感觉直接推广到可测函数上.
需要用到逆函数定理, 很有意思.
\]
Chapter 9 General concepts of Probability
Index:
- elementary event P131
- laws P131
- Random variable P133
- binomial law P138
- Characteristic function P139
注意:
\]
是限制在\(\mathbb{P}\)-integrable之上的.
Chapter 10 Conditional probability and independece
Index:
- Independece of two families P147
- \(\sigma\)-algebra generated by a random variable P147
- Independence of two random variables P147
- Independence of familes \(\mathscr{A}_i\) P149
- \(\sigma(X):= \{\{X \in A\}:A \in \mathscr{E}\}\) P149
- \(\sigma(\{X\}_{i \in I})\) P152
- independent and identically distributed P155
由条件概率衍生到独立性, 随机变量的独立性有几个等价条件P147, P150.
需要区分联合分布的概率和\(\mu\times v\)的区别 (当独立时才等价).
Chapter 11 Convergence of random variables
| 测度 | 概率 |
|---|---|
| 一致收敛 | 一致收敛 |
| 几乎一致收敛 | 几乎一致收敛 |
| 几乎处处收敛 | 几乎处处收敛 |
| 依测度收敛 | 依概率收敛 |
| \(L^p\)收敛 | \(\lim_{n\rightarrow \infty}\mathbb{E}(\cdot)^p=0\) |
| 弱收敛 | 依分布收敛 |
(几乎)一致收敛可以得到几乎处处和依测度收敛.
几乎处处在测度有限的情况下可以推几乎一致收敛, 从而得到依测度收敛.
依测度收敛必存在一个几乎处出收敛的子列.
\(L^p\)收敛一定能够有依测度收敛.
特别地, 依概率收敛有依分布收敛, 只有当依分布收敛到常数\(c\)的时候, 才能推依概率收敛到\(c\)(对应的有限测度).
Chapter 12 Sequences of independent variables
Index:
- terminal \(\sigma\)-algerba \(\cap_{n} \mathscr{B}_n\) P172
- empirical distribution function P180
Kolmogorov's dichotomy P173 很有趣.
大数定律再到中心极限定理.
Chapter 13 Stationary sequences and elements of ergodic theory
Index:
- stationary sequences P186
- measure-preserving transformation P188
- T-invariant P189
- Ergodic maps P189
- conjugate maps P190
平稳序列的定义需要注意, 另外一些理论有趣却渐渐脱离了掌控, 有点摸不着头脑.