实分析笔记

1.Lebesgue 积分的极限理论

1.1 Beppo Levi 定理

Beppo Levi 定理: 设 $E \subset \mathbb{R}^d$ 是 Lebesgue 可测集, $f_n, n\in \mathbb{N}$ 是 $E$ 上的非负递增的可测函数列, 且
$\lim_{n \to \infty}f_n(x)=f(x),a.e. x \in E$
则
$\lim_{n \to \infty}\int_{E}f_{n}dm=\int_{E}\lim_{n \to \infty}f_{n}dm.$
注: Beppo Levi was an Italian mathematician who wrote articles on logic, differential equations, complex variable, as well as on the border between analysis and physics.

Levi_Beppo_4.jpeg

1.2 Lebesgue 控制收敛

Lebesgue 逐项积分定理: 设 $E \subset \mathbb{R}^d$ 是 Lebesgue 可测集, $f_n, n\in \mathbb{N}$ 是 $E$ 上的非负的可测函数列,则
$\int_{E}\sum_{n=1}^{\infty}f_{n}dm=\sum_{n=1}^{\infty}\int_{E}f_{n}dm$
Lebesgue 控制收敛定理: 设 $E \subset \mathbb{R}^d$ 是 Lebesgue 可测集, $f, f_n, n\in \mathbb{N}$ 是 $E$ 上的可测函数列, 如果

在 $E$ 存在Lebesgue 可积函数, 使得 $|f_n(x)|\leq F(x),a.e. x \in E, \forall n \in \mathbb{N}$ .
$\lim_{n \to \infty }f_{n}(x)=f(x),a.e. x \in E$ .

则 $f, f_n, n\in \mathbb{N}$ 在 $E$ 上是 Lebesgue 可积分的, 且
$\lim_{n \to \infty}\int_{E}f_{n}dm=\int_{E}\lim_{n \to \infty}f_{n}dm=\int_{E}fdm$

注: Henri-Léon Lebesgue, (born June 28, 1875, Beauvais France—died July 26, 1941, Paris), French mathematician whose generalization of the Riemann integral revolutionized the field of integration

Henri-Léon Lebesgue.jpeg

推论: 设 $E \subset \mathbb{R}^d$ 是 Lebesgue 可测集, 非负函数 $f$ 在 $E$ 上 Lebesgue 可积, $E_k$ 是 $E$ 的一个可测分割, 则
$\int_{E}fdm=\sum_{n=1}^{\infty}\int_{E_n}fdm$

1.3 Fatou 引理

Fatou 引理: 设 $E \subset \mathbb{R}^d$ 是 Lebesgue 可测集, $f_n, n\in \mathbb{N}$ 是 $E$ 上的非负的可测函数列, 则
$\int_{E}\varliminf_{n \to \infty}f_{n}dm \leq \varliminf_{n \to \infty}\int_{E}f_{n}dm.$
注: Pierre Joseph Louis Fatou was a French mathematician and astronomer who worked in several branches of analysis.

Pierre Joseph Louis Fatou.jpeg

2. Fubini定理

Tonelli 定理: 设 $f$ 是 $\mathbb{R}^{p}\times \mathbb{R}^{q}$ 上非负 Lebesgue 可测函数, 则

对几乎所有的 $x \in \mathbb{R}^{p}$ , $f(x,\cdot)$ 是 $\mathbb{R}^{q}$ 上的非负Lebesgue 可测函数;
$\mathbb{R}^{p}$ 上的函数 $F_{f}(x)=\int_{\mathbb{R}^{p}}f(x,y)dy,\forall x \in \mathbb{R}^{p}$ 是非负Lebesgue 可测的.
$\int_{\mathbb{R}^{p+q}}f(x,y)dxdy=\int_{\mathbb{R}^{p}}\int_{\mathbb{R}^{q}}f(x,y)dy$

注: Leonida Tonelli was an Italian mathematician who worked on the theory of functions and the calculus of variations.

Leonida Tonelli.jpeg

Fubini 定理: 设 $f$ 是 $\mathbb{R}^{p}\times \mathbb{R}^{q}$ 上Lebesgue 可测函数, 则

对几乎所有的 $x \in \mathbb{R}^{p}$ , $f(x,\cdot)$ 是 $\mathbb{R}^{q}$ 上的Lebesgue 可积函数;
$\mathbb{R}^{p}$ 上的函数 $F_{f}(x)=\int_{\mathbb{R}^{p}}f(x,y)dy,\forall x \in \mathbb{R}^{p}$ 是Lebesgue 可积函数.
$\int_{\mathbb{R}^{p+q}}f(x,y)dxdy=\int_{\mathbb{R}^{p}}\int_{\mathbb{R}^{q}}f(x,y)dy$

注: Guido Fubini was an Italian mathematician who worked in many different areas including analysis, the calculus of variations and group theory.