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# 数学代写|数学分析作业代写Mathematical Analysis代考|MATH1013 The Riemann Integral

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## 数学代写|数学分析代写MATHEMATICAL ANALYSIS代考|The Riemann Integral

In this section, we treat the definition and the fundamental properties of the Riemann integral of a bounded function on a compact box. The main reason for the inclusion of this section is that our definition of Lebesgue measure is, loosely stated, based on the notion that the Riemann integral of a continuous function $f$ on a compact box measures the volume of the region below the graph of $f$. The presentation in this section is standard and reflects almost exactly the standard approach to the Riemann integral on a compact interval found in undergraduate real analysis textbooks.

Let $I=[a, b]$ be a compact interval. A grid in $I$ is a sequence of points $x_0=a<$ $x_1<x_2<\ldots<x_k=b$

Each grid in $I$ defines a partition of $I$ into a finite set of closed intervals $\mathcal{P}=\left{\left[x_0, x_1\right],\left[x_1, x_2\right], \ldots,\left[x_{k-1}, x_k\right]\right}$. We make no distinction between a grid in $I$ and the partition it generates. We also denote a partition of $I$ by the sequence that defines it, $\left{x_0, \ldots, x_k\right}$. We say that a partition $\mathcal{P}^{\prime}=\left{y_0, \ldots, y_m\right}$ is a refinement of a partition $\mathcal{P}=\left{x_0, \ldots, x_k\right}$ if $\left{x_0, \ldots, x_k\right} \subseteq\left{y_0, \ldots, y_m\right}$. This simply means that $\mathcal{P}^{\prime}$ is obtained from $\mathcal{P}$ by inserting additional grid points between some (or all) consecutive points $x_i$ and $x_{i+1}$. Note that if $\mathcal{P}^{\prime}$ is a refinement of $\mathcal{P}$, then every interval in $\mathcal{P}$ is the union of intervals in $\mathcal{P}^{\prime}$. If $\mathcal{P}$ and $\mathcal{P}^{\prime}$ are partitions of $[a, b]$, then $\mathcal{P}$ and $\mathcal{P}^{\prime}$ have a common refinement, namely, the partition generated by the $\operatorname{grid}\left{x_0, \ldots, x_k\right} \cup\left{y_0, \ldots, y_m\right}$

## 数学代写|数学分析代写MATHEMATICAL ANALYSIS代考|Measure Spaces

Let us consider the problem of measuring the volume of objects (sets) in $\mathbb{R}^3$. Strictly speaking, volume is a function that assigns a nonnegative number to a subset of $\mathbb{R}^3$. A natural question is whether it is possible to measure the volume of an arbitrary subset of $\mathbb{R}^3$. For the most natural measure on $\mathbb{R}^3$, namely, the Lebesgue measure, the answer to the question is no. In other words, there are subsets of $\mathbb{R}^3$ to which a volume cannot be assigned. The question then becomes that of finding a large enough collection of $\mathbb{R}^3$ for which a volume can be assigned. Such sets are called measurable. It is clearly desirable for the finite union of measurable sets to be measurable. It was a paradigm shift when it was realized that a successful formulation of a measure theory necessitates that we allow the countable union of measurable sets to be measurable, and this leads to the definition of a $\sigma$-algebra. The definition of a measure as a set function on a $\sigma$-algebra is quite intuitive. This section develops the basics of abstract measure theory and measurable functions. The picture continues to evolve and culminates in section $8.4$ with the construction of the Lebesgue measure.

For the remainder of this chapter, we use the notation $E^{\prime}$ for the complement $X-E$ of a subset $E$ of a set $X$.

Definitions. A collection $\mathfrak{M}$ of subsets of a nonempty set $X$ is said to be an algebra of sets in $X$ if the following two conditions are met:
(a) if $E \in \mathfrak{M}$, then $E^{\prime} \in \mathfrak{M}$; and
(b) if $E_1, E_2 \in \mathfrak{M}$, then $E_1 \cup E_2 \in \mathfrak{M}$.
An algebra $\mathfrak{M}$ is called a $\sigma$-algebra if it satisfies the additional condition
(c) if $\left(E_n\right)$ is a sequence in $\mathfrak{M}$, then $\cup_{n=1}^{\infty} E_n \in \mathfrak{M}$.

## 数学代写|数学分析代写MATHEMATICAL ANALYSIS代考|The Riemann Integral

.我们不区分网格I及其生成

\left 缺少或无法识别的分隔符 $\quad$ 是分区的细化left 缺少或无法识别的分隔符 $\quad$ 如果
\left 缺少或无法识别的分隔符 $\quad$. 这仅仅意味着 $\mathcal{P}^{\prime}$ 是从 $\mathcal{P}{\text {通过在一些 (或所有) 连续点之间揷入额外的网格点 } x_i}$ 和 $x{i+1}$. 请暗意，如果 $\mathcal{P}^{\prime}$ 是对 $\mathcal{P}$, 那 $/$ 每个区间在 $\mathcal{P}$ 是区间的联合 $\mathcal{P}^{\prime}$. 如果 $\mathcal{P}$ 和 $\mathcal{P}^{\prime}$ 是分区 $[a, b] ＼mathrm{~ ， 然 后 ~} \mathcal{P}$ 和 $\mathcal{P}^{\prime}$ 有一个共同的细化， 即由》left 缺少或无法识别的分隔符

## 数学代写数学分析代写MATHEMATICAL ANALYSIS代考|Measure Spaces

(a) 如果 $E \in \mathfrak{M}$ ，然后 $E^{\prime} \in \mathfrak{M} ;($
b) 如果 $E_1, E_2 \in \mathfrak{M}$ ，然后 $E_1 \cup E_2 \in \mathfrak{M}$.

(c) 如果 $\left(E_n\right)$ 是一个序列 $\mathfrak{M}$ ，然后 $\cup_{n=1}^{\infty} E_n \in \mathfrak{M}$.

## 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。