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# 数学代写|勒贝格积分代写Lebesgue Integration代考|BSMA2003 Measurable Functions

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## 数学代写|勒贝格积分代写Lebesgue Integration代考|Measurable Functions

In this chapter we want to define the Lebesgue integral in a fashion which is analogous to our definitions of regulated integral and Riemann integral from Chapter 1. The difference is that we will no longer use step functions to approximate a function we want to integrate, but instead will use a much more general class called simple functions.
Definition 3.1.1. (Characteristic function). If $A \subset[0,1]$, its characteristic function $\mathfrak{X}A(x)$ (sometimes called the indicator function) is defined by $$\mathfrak{X}_A(x)= \begin{cases}1, & \text { if } x \in A ; \ 0, & \text { otherwise } .\end{cases}$$ Definition 3.1.2. (Measurable partition). A finite measurable partition of $[0,1]$ is a collection $\left{A_i\right}{i=1}^n$ of measurable subsets which are pairwise disjoint and whose union is $[0,1]$.

We can now define simple functions. Like step functions these functions have only finitely many values, but unlike step functions the set on which a simple function assumes a given value is no longer an interval. Instead, a simple function is constant on each subset of a finite measurable partition of $[0,1]$.

## 数学代写|勒贝格积分代写Lebesgue Integration代考|The Lebesgue Integral of Bounded Functions

In this section we want to define the Lebesgue integral and characterize the bounded integrable functions. In the case of the regulated integral, the integrable functions are the uniform limits of step functions. In the case of the Riemann integral a function $f$ is integrable if the infimum of the integrals of step functions larger than $f$ equals the supremum of the integrals of step functions less than $f$. It is natural to alter both of these definitions, replacing step functions with simple functions. It turns out that when we do this for bounded functions we get the same class of integrable functions whether we use the analog of regulated integral or the analog of Riemann integral. Moreover, this class is precisely the bounded measurable functions!

Theorem 3.2.1. If $f:[0,1] \rightarrow \mathbb{R}$ is a bounded function, then the following are equivalent:
(1) The function $f$ is Lebesgue measurable.
(2) There is a sequence of simple functions $\left{f_n\right}_{n=1}^{\infty}$ which converges uniformly to $f$.
(3) If $\mathcal{U}\mu(f)$ denotes the set of all simple functions $u(x)$ such that $f(x) \leq u(x)$ for all $x$ and if $\mathcal{L}\mu(f)$ denotes the set of all simple functions $v(x)$ such that $v(x) \leq f(x)$ for all $x$, then
$$\sup {v \in \mathcal{L}\mu(f)}\left{\int v d \mu\right}=\inf {u \in \mathcal{U}\mu(f)}\left{\int u d \mu\right}$$

## 数学代写|勒贝格积分代写|勒贝格积分代考|可测量函数

$mathfrak{X}_A(x)={1, \quad$ 如果$x\在A中; 0, \quad$ 否则。

## 数学代写勒贝格积分代写Lebesgue Integration代写|The Lebesgue Integral of Bounded Functions

(1) 函数$f$是Lebesgue可测的。
(2) 存在一个简单的函数序列 缺少或不被认可的分界符的$left，它均匀地收敛于$f$。 (3) 如果$/mathcal{U} \表示所有简单函数$u(x)$的集合，使得$f(x)\leq u(x)$对于所有$x$，并且如果$mathcal{L} \表示所有简单函数$v(x)$的集合，使得$v(x)\leq f(x)$适用于所有$x\$，那么

## 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。
The provided text is a part of a mathematics textbook or lecture notes on Lebesgue Integration. It introduces the concept of measurable functions and simple functions, and then discusses the definition of the Lebesgue integral for bounded functions.
no_question

the text does not contain any mathematical questions or problems to solve. it is purely expository, providing definitions and theorems related to lebesgue integration.
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
Posted on Categories:Lebesgue Integration, 勒贝格积分, 数学代写

# 数学代写|勒贝格积分代写Lebesgue Integration代考|BSMA2003 Measurable Functions

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## 数学代写|勒贝格积分代写Lebesgue Integration代考|Measurable Functions

In this chapter we want to define the Lebesgue integral in a fashion which is analogous to our definitions of regulated integral and Riemann integral from Chapter 1. The difference is that we will no longer use step functions to approximate a function we want to integrate, but instead will use a much more general class called simple functions.
Definition 3.1.1. (Characteristic function). If $A \subset[0,1]$, its characteristic function $\mathfrak{X}A(x)$ (sometimes called the indicator function) is defined by $$\mathfrak{X}_A(x)= \begin{cases}1, & \text { if } x \in A ; \ 0, & \text { otherwise } .\end{cases}$$ Definition 3.1.2. (Measurable partition). A finite measurable partition of $[0,1]$ is a collection $\left{A_i\right}{i=1}^n$ of measurable subsets which are pairwise disjoint and whose union is $[0,1]$.

We can now define simple functions. Like step functions these functions have only finitely many values, but unlike step functions the set on which a simple function assumes a given value is no longer an interval. Instead, a simple function is constant on each subset of a finite measurable partition of $[0,1]$.

## 数学代写|勒贝格积分代写Lebesgue Integration代考|The Lebesgue Integral of Bounded Functions

In this section we want to define the Lebesgue integral and characterize the bounded integrable functions. In the case of the regulated integral, the integrable functions are the uniform limits of step functions. In the case of the Riemann integral a function $f$ is integrable if the infimum of the integrals of step functions larger than $f$ equals the supremum of the integrals of step functions less than $f$. It is natural to alter both of these definitions, replacing step functions with simple functions. It turns out that when we do this for bounded functions we get the same class of integrable functions whether we use the analog of regulated integral or the analog of Riemann integral. Moreover, this class is precisely the bounded measurable functions!

Theorem 3.2.1. If $f:[0,1] \rightarrow \mathbb{R}$ is a bounded function, then the following are equivalent:
(1) The function $f$ is Lebesgue measurable.
(2) There is a sequence of simple functions $\left{f_n\right}_{n=1}^{\infty}$ which converges uniformly to $f$.
(3) If $\mathcal{U}\mu(f)$ denotes the set of all simple functions $u(x)$ such that $f(x) \leq u(x)$ for all $x$ and if $\mathcal{L}\mu(f)$ denotes the set of all simple functions $v(x)$ such that $v(x) \leq f(x)$ for all $x$, then
$$\sup {v \in \mathcal{L}\mu(f)}\left{\int v d \mu\right}=\inf {u \in \mathcal{U}\mu(f)}\left{\int u d \mu\right}$$

## 数学代写|勒贝格积分代写|勒贝格积分代考|可测量函数

$mathfrak{X}_A(x)={1, \quad$ 如果$x\在A中; 0, \quad$ 否则。

## 数学代写勒贝格积分代写Lebesgue Integration代写|The Lebesgue Integral of Bounded Functions

(1) 函数$f$是Lebesgue可测的。
(2) 存在一个简单的函数序列 缺少或不被认可的分界符的$left，它均匀地收敛于$f$。 (3) 如果$/mathcal{U} \表示所有简单函数$u(x)$的集合，使得$f(x)\leq u(x)$对于所有$x$，并且如果$mathcal{L} \表示所有简单函数$v(x)$的集合，使得$v(x)\leq f(x)$适用于所有$x\$，那么

## 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。
user
The provided text is a part of a mathematics textbook or lecture notes on Lebesgue Integration. It introduces the concept of measurable functions and simple functions, and then discusses the definition of the Lebesgue integral for bounded functions.

TYPE: NO_QUESTION

The text does not contain any mathematical questions or problems to solve. It is purely expository, providing definitions and theorems related to Lebesgue integration.