An Introduction to Measure Theory and Probability

目录

• 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

\[\varphi(x)=\sum_{k-1}^n a_k 1_{A_k}, A_k = \varphi^{-1}(\{a_k\}).
\]

• 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

\[\int_X \varphi d\mu := \int_{0}^{\infty} \mu(\{\varphi > t\}) \mathrm{d}t,
\]

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

\[f(x)=f(a)+\int_a^x g(t)\mathrm{d}t,
\]

\[\lim_{r\downarrow0} \frac{1}{\omega_n r^n} \int_{B_r(x)} |f(y)-f(x)|\mathrm{d}y=0.
\]

Chapter 8 Measurable transformations

Index:

• differential P123
• Jacobian determinant P125
• diffeomorphism P125
• critical set \(C_F\) P125

\[F_\# \mu(I) := \mu(F^{-1}(I))
\]

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.

\[\int_{F(U)} \varphi(y) \mathrm{d}y = \int_{U} \varphi(F(x)) |JF|(x)\mathrm{d}x.
\]

Chapter 9 General concepts of Probability

Index:

• elementary event P131
• laws P131
• Random variable P133
• binomial law P138
• Characteristic function P139

注意:

\[\mathbb{E}_{\mathbb{P}}(X):= \int_{\Omega} X(\omega) \mathrm{d} \mathbb{P}(\omega),
\]

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.