# 数学代写|数学分析代写Mathematical Analysis代考|Abstract Integration

#### Doug I. Jones

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## 数学代写|数学分析代写Mathematical Analysis代考|Abstract Integration

In this section, we examine Lebesgue’s revolutionary approach to the definition of the integral. The motivation below is imprecise and does not rigorously develop any particular set of ideas. For the sake of simplicity, we assume that $f$ is a positive continuous function on a compact interval.

The Riemann integral is based on the simple geometrical idea of dividing the region below the graph of $f$ into thin vertical strips, where the area below the graph is approximated by the integral of a step function (the Riemann sum). Lebesgue’s idea was to divide the range of $f$ by points $y_0, \ldots, y_n$, and, for $k=1, \ldots, n$, we consider the sets $E_k=f^{-1}\left(\left[y_k, y_{k+1}\right)\right)$. Even for an uncomplicated function, the set $E_k$ may come in several fragments, as shown in figure 8.1 , where $E_k$ has three fragments. When $y_{k+1}-y_k$ is small, the approximate combined area of the three shaded strips is the approximate common height, $y_k$, times the sum of the lengths of the three fragments that comprise the set $E_k$ or, more precisely, the measure of $E_k$. Thus the approximate area below the graph is $\sum_{k=1}^n y_k \mu\left(E_k\right)$, which, by definition, is the integral of a simple function. Needless to say, as the partition of the range of $f$ gets finer, we expect the integrals of the simple functions to converge to the integral of $f$. This is the overarching idea in Lebesgue integration. As it turns out, we can integrate far more functions under the Lebesgue definition than under the Riemann definition. For example, the integral of any positive measurable function is defined, although it may not be finite. Additionally, the definition of the integral extends seamlessly to abstract measure spaces. The section results capture the above ideas. First we define the integral of a positive measurable function $f$, then we show that $f$ is the limit of simple functions, $s_n$, and then we show that $\int_X f d \mu=\lim _n \int_X s_n d \mu$. Extending the definition of the integral to complex functions follows without difficulty. The section concludes with three important convergence theorems.

## 数学代写|数学分析代写Mathematical Analysis代考|Convergence Theorems

Theorem 8.3.8 (Fatou’s theorem). Let $f_n: X \rightarrow[0, \infty]$ be a sequence of measurable functions. Then
$$\int_X \liminf n f_n d \mu \leq \liminf _n \int_X f_n d \mu .$$ Proof. Let $g_n=i n f{k \geq n} f_k$. Then $0 \leq g_1 \leq g_2 \leq \ldots$, and let $f(x)=\lim _n g_n(x)$. Note that $f(x)=\liminf _n f_n(x)$. If s is a simple function such that $0 \leq s \leq f$, then, by lemma 8.3.3, $\int_X s d \mu \leq \lim _n \int_X g_n d \mu$. Hence $\int_X f d \mu=\sup \left{\int_X s d \mu: s \leq f\right} \leq \lim _n \int_X g_n d \mu$. Since $g_n \leq f_n, \int_X g_n d \mu \leq \int_X f_n d \mu$, and $\lim _n \int_X g_n d \mu \leq \liminf _n \int_X f_n d \mu$.

Example 5. Let $\left(f_n\right)$ be a convergent sequence in $\mathbf{Q}^{\mathbf{1}}(\mu)$, and let $f$ be its $\mathbf{Q}^{\mathbf{1}}$-limit. Then $\left(f_n\right)$ contains a subsequence that converges to $f$ for almost every $x \in X$.
Choose a subsequence $\left(f_{n_i}\right)$ of $\left(f_n\right)$ such that, for $i \in \mathbb{N},\left|f_{n_i}-f\right|_1<2^{-i}$. Let $g_k=\sum_{i=1}^k\left|f_{n_i}-f\right|$. The functions $g_k$ are in $\mathfrak{Q}^1$ and, by construction, $0 \leq g_1 \leq$ $g_2 \leq \ldots$, and $\left|g_k\right|_1 \leq 1$. Let $g(x)=\lim k g_k(x)$. By Fatou’s theorem, $\int_X g d \mu \leq$ $\liminf _n\left|g_k\right|_1 \leq 1$. This shows that $g \in \mathfrak{2}^1$. $^5$ Since $g(x)=\sum{i=1}^{\infty}\left|f_{n_i}(x)-f(x)\right|$, it follows that the series $\sum_{i=1}^{\infty}\left|f_{n_i}(x)-f(x)\right|$ is convergent for a.e. $x \in X$ (by example 1). In particular, $\lim {i \rightarrow \infty}\left|f{n_i}(x)-f(x)\right|=0$ for a.e. $x \in X$.

# 数学分析代考

## 数学代写|数学分析代写Mathematical Analysis代考|Convergence Theorems

$$\int_X \liminf n f_n d \mu \leq \liminf _n \int_X f_n d \mu .$$证明。让$g_n=i n f{k \geq n} f_k$。然后$0 \leq g_1 \leq g_2 \leq \ldots$，让$f(x)=\lim _n g_n(x)$。请注意$f(x)=\liminf _n f_n(x)$。如果s是一个简单函数，满足$0 \leq s \leq f$，那么，根据引理8.3.3,$\int_X s d \mu \leq \lim _n \int_X g_n d \mu$。因此，$\int_X f d \mu=\sup \left{\int_X s d \mu: s \leq f\right} \leq \lim _n \int_X g_n d \mu$。自从$g_n \leq f_n, \int_X g_n d \mu \leq \int_X f_n d \mu$和$\lim _n \int_X g_n d \mu \leq \liminf _n \int_X f_n d \mu$。

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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