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 dmu := 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 ( ilde{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 ightarrow 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_{rdownarrow0} 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 imes 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 ightarrow 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.