An Introduction to Measure Theory and Probability

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.