## 数学代写|傅里叶分析代写Fourier analysis代考|MATH4303

statistics-lab™ 为您的留学生涯保驾护航 在代写傅里叶分析Fourier analysis方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写傅里叶分析Fourier analysis代写方面经验极为丰富，各种代写傅里叶分析Fourier analysis相关的作业也就用不着说。

## 数学代写|傅里叶分析代写Fourier analysis代考|Integral Formulas

We will be using a number of integral formulas in our work, most of which should be well known from basic calculus. For example, you surely recall that no one really calculates an integral via Riemann sums. Instead, we use the fact that, as long as $f$ is uniformly smooth on a finite interval $(\alpha, \beta)$,
$$\int_\alpha^\beta f^{\prime}(x) d x=” f(\beta)-f(\alpha) ” .$$
Notice the quotes around the right-hand side of this equation. As written, this formula assumes $f$ is continuous at the endpoints of $(\alpha, \beta)$. Often, though, we will be dealing with functions that have jump discontinuities at the endpoints of the intervals over which we are integrating. In these cases, the correct formula is actually
$$\int_\alpha^\beta f^{\prime}(x) d x=\lim {x \rightarrow \beta^{-}} f(x)-\lim {x \rightarrow \alpha^{+}} f(x) .$$
For convenience, this will often be written as
$$\int_\alpha^\beta f^{\prime}(x) d x=\left.f(x)\right|\alpha ^\beta,$$ where it is understood that $$\left.f(x)\right|\alpha ^\beta=\lim {x \rightarrow \beta^{-}} f(x)-\lim {x \rightarrow \alpha^{+}} f(x) .$$
Because we will often be integrating functions that are not smooth, let us state and verify the following slight generalization of the above:
Theorem 4.1
Let $f$ be continuous and piecewise smooth on the finite interval $(\alpha, \beta)$. Then
$$\int_\alpha^\beta f^{\prime}(x) d x=\left.f(x)\right|_\alpha ^\beta$$

PROOF (partial): First of all, if $f^{\prime}$ has no discontinuities, then $f$ is uniformly smooth on $(\alpha, \beta)$ and, from elementary calculus, we know equation (4.7) holds.

If $f^{\prime}$ has only one discontinuity in $(\alpha, \beta)$, say, at $x=x_0$, then $f$ is uniformly smooth on $\left(\alpha, x_0\right)$ and $\left(x_0, \beta\right)$. Thus,
\begin{aligned} \int_\alpha^\beta f^{\prime}(x) d x & =\int_\alpha^{x_0} f^{\prime}(x) d x+\int_{x_0}^\beta f^{\prime}(x) d x \ & =\left[\lim {x \rightarrow x_0^{-}} f(x)-\lim {x \rightarrow \alpha^{+}} f(x)\right]+\left[\lim {x \rightarrow \beta-} f(x)-\lim {x \rightarrow x_0^{+}} f(x)\right] \ & =\lim {x \rightarrow \beta^{-}} f(x)-\lim {x \rightarrow \alpha^{+}} f(x)+\lim {x \rightarrow x_0^{-}} f(x)-\lim {x \rightarrow x_0^{+}} f(x) . \end{aligned}

## 数学代写|傅里叶分析代写Fourier analysis代考|Infinite Series (Summations)

For mathematicians (and others indoctrinated by mathematicians – like you), an infinite series is simply any expression that looks like the summation of an infinite number of things. For example, you should recognize
$$\sum_{k=1}^{\infty} \frac{1}{k}=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\cdots$$
(with the “…” denoting “continue the obvious pattern”) as the famous harmonic series.
In Fourier analysis we must deal with infinite series of numbers, infinite series of functions, and, ultimately, infinite series of generalized functions. Here, we will review some basic facts concerning infinite series of numbers. Later, as the need arises, we’ll extend our discussions to include those other infinite series.
Basic Facts
Let $c_0, c_1, c_2, \ldots$ be any sequence of numbers, and consider the infinite series with these numbers as its terms,
$$\sum_{k=0}^{\infty} c_k=c_0+c_1+c_2+\cdots .$$
Here the index, $k$, started at 0 . In practice, it can start at any convenient integer $M$. For any integer $N$ with $N \geq 0$ (or, more generally, with $N \geq M$ ), the $N^{\text {th }}$ partial sum $S_N$ is simply the value obtained by adding all the terms up to and including $c_N$,
$$S_N=\sum_{k=0}^N c_k=c_0+c_1+c_2+\cdots+c_N .$$
The sum (or value) of the infinite series, which is also denoted by $\sum_{k=0}^{\infty} c_k$, is the value we get by taking the limit of the $N^{\text {th }}$ partial sums as $N \rightarrow \infty$,
$$\sum_{k=0}^{\infty} c_k=\lim {N \rightarrow \infty} S_N=\lim {N \rightarrow \infty} \sum_{k=0}^N c_k .$$

# 傅里叶分析代写

## 数学代写|傅里叶分析代写Fourier analysis代考|Integral Formulas

$$\int_\alpha^\beta f^{\prime}(x) d x=” f(\beta)-f(\alpha) ” .$$

$$\int_\alpha^\beta f^{\prime}(x) d x=\lim {x \rightarrow \beta^{-}} f(x)-\lim {x \rightarrow \alpha^{+}} f(x) .$$

$$\int_\alpha^\beta f^{\prime}(x) d x=\left.f(x)\right|\alpha ^\beta,$$其中，$$\left.f(x)\right|\alpha ^\beta=\lim {x \rightarrow \beta^{-}} f(x)-\lim {x \rightarrow \alpha^{+}} f(x) .$$

$$\int_\alpha^\beta f^{\prime}(x) d x=\left.f(x)\right|_\alpha ^\beta$$

\begin{aligned} \int_\alpha^\beta f^{\prime}(x) d x & =\int_\alpha^{x_0} f^{\prime}(x) d x+\int_{x_0}^\beta f^{\prime}(x) d x \ & =\left[\lim {x \rightarrow x_0^{-}} f(x)-\lim {x \rightarrow \alpha^{+}} f(x)\right]+\left[\lim {x \rightarrow \beta-} f(x)-\lim {x \rightarrow x_0^{+}} f(x)\right] \ & =\lim {x \rightarrow \beta^{-}} f(x)-\lim {x \rightarrow \alpha^{+}} f(x)+\lim {x \rightarrow x_0^{-}} f(x)-\lim {x \rightarrow x_0^{+}} f(x) . \end{aligned}

## 数学代写|傅里叶分析代写Fourier analysis代考|Infinite Series (Summations)

$$\sum_{k=1}^{\infty} \frac{1}{k}=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\cdots$$
(“…”表示“继续明显的模式”)作为著名的和声级数。

$$\sum_{k=0}^{\infty} c_k=c_0+c_1+c_2+\cdots .$$

$$S_N=\sum_{k=0}^N c_k=c_0+c_1+c_2+\cdots+c_N .$$

$$\sum_{k=0}^{\infty} c_k=\lim {N \rightarrow \infty} S_N=\lim {N \rightarrow \infty} \sum_{k=0}^N c_k .$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|傅里叶分析代写Fourier analysis代考|MATH668

statistics-lab™ 为您的留学生涯保驾护航 在代写傅里叶分析Fourier analysis方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写傅里叶分析Fourier analysis代写方面经验极为丰富，各种代写傅里叶分析Fourier analysis相关的作业也就用不着说。

## 数学代写|傅里叶分析代写Fourier analysis代考|Some Useful Inequalities

At various points in our work we will need to determine “suitable upper bounds” for various numerical expressions. At some of these points, the inequalities discussed below will be invaluable.

Two basic inequalities will be identified. You are probably well acquainted with the first one, the triangle inequality, though you may not have given it a name before. You may not be as well acquainted with the second one, the Schwarz inequality. It is somewhat more subtle than the triangle inequality and will require a formal proof. Both, it should be mentioned, are fundamental inequalities in analysis and have applications and generalizations beyond the simple formulas discussed in this section.
The Triangle Inequality
Let $A$ and $B$ be any two real numbers. If you just consider how values of $|A|,|B|,|A+B|$, and $|A|+|B|$ depend on the signs of $A$ and $B$, then you should realize that
$$|A+B| \leq|A|+|B|$$
This inequality is called the triangle inequality. The reason for its name is explained in chapter 6 (see page 58), where it is also shown that this inequality holds when $A$ and $B$ are complex numbers as well.

There are two other inequalities that we can immediately derive from the triangle inequality. The first is the obvious extension to the case where we are adding up some (finite) set of numbers $\left{A_1, A_2, A_3, \ldots, A_N\right}$. Successively applying the triangle inequality,
\begin{aligned} \left|A_1+A_2+A_3+\cdots+A_N\right| & \leq\left|A_1\right|+\left|A_2+A_3+\cdots+A_N\right| \ & \leq\left|A_1\right|+\left|A_2\right|+\left|A_3+\cdots+A_N\right| \ & \leq \cdots, \end{aligned}
we are, eventually, left with the inequality
$$\left|A_1+A_2+A_3+\cdots+A_N\right| \leq\left|A_1\right|+\left|A_2\right|+\left|A_3\right|+\cdots+\left|A_N\right|,$$
which can also be called the triangle inequality.

## 数学代写|傅里叶分析代写Fourier analysis代考|The Schwarz Inequality (for Finite Sums)

The Schwarz inequality is a generalization of the well-known fact that, if $\mathbf{a}$ and $\mathbf{b}$ are any two two- or three-dimensional vectors, then
$$|\mathbf{a} \cdot \mathbf{b}| \leq|\mathbf{a}||\mathbf{b}|$$
In component form, with $\mathbf{a}=\left(a_1, a_2, a_3\right)$ and $\mathbf{b}=\left(b_1, b_2, b_3\right)$, this inequality is
$$\left|\sum_{k=1}^3 a_k b_k\right| \leq\left(\sum_{k=1}^3\left|a_k\right|^2\right)^{1 / 2}\left(\sum_{k=1}^3\left|b_k\right|^2\right)^{1 / 2} .$$

Theorem 3.7 (Schwarz inequality for finite summations)
Let $N$ be any integer, and let $\left{a_1, a_2, a_3, \ldots, a_N\right}$ and $\left{b_1, b_2, b_3, \ldots, b_N\right}$ be any two sets of $N$ numbers (real or complex). Then,
$$\left|\sum_{k=1}^N a_k b_k\right| \leq\left(\sum_{k=1}^N\left|a_k\right|^2\right)^{1 / 2}\left(\sum_{k=1}^N\left|b_k\right|^2\right)^{1 / 2} .$$
PROOF: Suppose we can show
$$\sum_{k=1}^N\left|a_k\right|\left|b_k\right| \leq\left(\sum_{k=1}^N\left|a_k\right|^2\right)^{1 / 2}\left(\sum_{k=1}^N\left|b_k\right|^2\right)^{1 / 2} .$$
Then inequality (3.7) follows immediately by combining the above inequality with the triangle inequality,
$$\left|\sum_{k=1}^N a_k b_k\right| \leq \sum_{k=1}^N\left|a_k b_k\right|=\sum_{k=1}^N\left|a_k\right|\left|b_k\right| .$$
So we only need to verify that inequality (3.8) holds.
Consider, first, the trivial case where either
$$\sum_{k=1}^N\left|a_k\right|^2=0 \quad \text { or } \quad \sum_{k=1}^N\left|b_k\right|^2=0$$

# 傅里叶分析代写

## 数学代写|傅里叶分析代写Fourier analysis代考|Some Useful Inequalities

$$|A+B| \leq|A|+|B|$$

\begin{aligned} \left|A_1+A_2+A_3+\cdots+A_N\right| & \leq\left|A_1\right|+\left|A_2+A_3+\cdots+A_N\right| \ & \leq\left|A_1\right|+\left|A_2\right|+\left|A_3+\cdots+A_N\right| \ & \leq \cdots, \end{aligned}

$$\left|A_1+A_2+A_3+\cdots+A_N\right| \leq\left|A_1\right|+\left|A_2\right|+\left|A_3\right|+\cdots+\left|A_N\right|,$$

## 数学代写|傅里叶分析代写Fourier analysis代考|The Schwarz Inequality (for Finite Sums)

Schwarz不等式是一个众所周知的事实的推广，如果$\mathbf{a}$和$\mathbf{b}$是任意两个二维或三维向量，则
$$|\mathbf{a} \cdot \mathbf{b}| \leq|\mathbf{a}||\mathbf{b}|$$

$$\left|\sum_{k=1}^3 a_k b_k\right| \leq\left(\sum_{k=1}^3\left|a_k\right|^2\right)^{1 / 2}\left(\sum_{k=1}^3\left|b_k\right|^2\right)^{1 / 2} .$$

$$\left|\sum_{k=1}^N a_k b_k\right| \leq\left(\sum_{k=1}^N\left|a_k\right|^2\right)^{1 / 2}\left(\sum_{k=1}^N\left|b_k\right|^2\right)^{1 / 2} .$$

$$\sum_{k=1}^N\left|a_k\right|\left|b_k\right| \leq\left(\sum_{k=1}^N\left|a_k\right|^2\right)^{1 / 2}\left(\sum_{k=1}^N\left|b_k\right|^2\right)^{1 / 2} .$$

$$\left|\sum_{k=1}^N a_k b_k\right| \leq \sum_{k=1}^N\left|a_k b_k\right|=\sum_{k=1}^N\left|a_k\right|\left|b_k\right| .$$

$$\sum_{k=1}^N\left|a_k\right|^2=0 \quad \text { or } \quad \sum_{k=1}^N\left|b_k\right|^2=0$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|傅里叶分析代写Fourier analysis代考|MATH382

statistics-lab™ 为您的留学生涯保驾护航 在代写傅里叶分析Fourier analysis方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写傅里叶分析Fourier analysis代写方面经验极为丰富，各种代写傅里叶分析Fourier analysis相关的作业也就用不着说。

## 数学代写|傅里叶分析代写Fourier analysis代考|Classifying Functions Based on Continuity Continuous Functions

A function $f$ is continuous on an interval $(\alpha, \beta)$ if and only if is continuous at each point in the interval. Remember that, if any finite subinterval of $(\alpha, \beta)$ contains a finite (but not infinite ${ }^4$ ) number of trivial discontinuities, then all trivial discontinuities are automatically assumed to have been removed.

$$f(x)=\frac{\sin (2 \pi x)}{\sin (\pi x)},$$
is continuous on the real line.
Even though a function is continuous on a given interval, it might still be rather poorly behaved near an endpoint of the interval. For example, even though the function $1 / x$ is continuous on the finite interval $(0,1)$, it is not bounded. Instead, it “blows up” around $x=0$. To exclude such functions from discussion when $(\alpha, \beta)$ is a finite interval, we will impose the condition of “uniform continuity”, as defined in the next paragraph.

Let $(\alpha, \beta)$ be a finite interval. The function $f$ is uniformly continuous on $(\alpha, \beta)$ if, in addition to being continuous on $(\alpha, \beta)$, its one-sided limits at the endpoints,
$$\lim {x \rightarrow \alpha^{+}} f(x) \quad \text { and } \quad \lim {x \rightarrow \beta^{-}} f(x) \quad,$$
both exist.

## 数学代写|傅里叶分析代写Fourier analysis代考|Discontinuous Functions

Fourier analysis would be of very limited value if it only dealt with continuous functions. Still, we won’t be able to deal with every possible discontinuous function. We will have to restrict our attention to discontinuous functions we can reasonably handle. Typically, the minimal continuity requirement that we can conveniently get away with is “piecewise continuity” over the interval of interest. Occasionally the requirements can be weakened so that we can deal with some functions that are merely “continuous over some partitioning of the interval”.
Because it is the more important, we will describe “piecewise continuity” first.
Let $f$ be a function defined on an interval $(\alpha, \beta)$. If $(\alpha, \beta)$ is a finite interval, then we will say $f$ is piecewise continuous on $(\alpha, \beta)$ if and only if all of the following three statements hold:

1. $f$ has at most a finite number (possibly zero) of discontinuities on $(\alpha, \beta)$.
2. All of the (nontrivial) discontinuities of $f$ on $(\alpha, \beta)$ are jump discontinuities.
3. Both $\lim {x \rightarrow \alpha^{+}} f(x)$ and $\lim {x \rightarrow \beta^{-}} f(x)$ exist (as finite numbers).
If, on the other hand, $(\alpha, \beta)$ is an infinite interval, then $f$ will be referred to as piecewise continuous on $(\alpha, \beta)$ if and only if it is piecewise continuous on each finite subinterval of $(\alpha, \beta)$.

It is important to realize that a piecewise continuous function is not simply “continuous over pieces of $(\alpha, \beta)$ “. To see this, let $(\alpha, \beta)$ be a finite interval, and let $x_1, x_2, \ldots, x_N$ be the points in $(\alpha, \beta)-$ indexed so that $x_1<x_2<\cdots<x_N-$ at which a given piecewise continuous function $f$ is discontinuous. These points partition $(\alpha, \beta)$ into a finite number of subintervals
$$\begin{array}{lllllll} \left.\alpha, x_1\right) & , & \left(x_1, x_2\right) & , & \left(x_2, x_3\right) & , & \ldots \end{array} \quad, \quad\left(x_N, \beta\right) \quad,$$
with $f$ being continuous over each of these subintervals. But the second and third parts of the definition also ensure that
$$\lim {x \rightarrow \alpha^{+}} f(x) \quad, \quad \lim {x \rightarrow x_1^{-}} f(x) \quad, \quad \lim {x \rightarrow x_1^{+}} f(x) \quad, \quad \lim {x \rightarrow x_2^{-}} f(x) \quad, \quad \ldots \quad, \quad \lim _{x \rightarrow \beta^{-}} f(x)$$
all exist (and are finite). Thus, not only is $f$ continuous on each of the above subintervals, it is uniformly continuous on each of the above subintervals. ${ }^5$

# 傅里叶分析代写

## 考|Classifying Functions Based on Continuity Continuous Functions

$$f(x)=\frac{\sin (2 \pi x)}{\sin (\pi x)},$$

$$\lim {x \rightarrow \alpha^{+}} f(x) \quad \text { and } \quad \lim {x \rightarrow \beta^{-}} f(x) \quad,$$

## 数学代写|傅里叶分析代写Fourier analysis代考|Discontinuous Functions

$f$ 在$(\alpha, \beta)$上最多有有限个不连续点(可能为零)。

$(\alpha, \beta)$上$f$的所有(非平凡)不连续都是跳变不连续。

$\lim {x \rightarrow \alpha^{+}} f(x)$和$\lim {x \rightarrow \beta^{-}} f(x)$都存在(作为有限的数字)。

$$\begin{array}{lllllll} \left.\alpha, x_1\right) & , & \left(x_1, x_2\right) & , & \left(x_2, x_3\right) & , & \ldots \end{array} \quad, \quad\left(x_N, \beta\right) \quad,$$

$$\lim {x \rightarrow \alpha^{+}} f(x) \quad, \quad \lim {x \rightarrow x_1^{-}} f(x) \quad, \quad \lim {x \rightarrow x_1^{+}} f(x) \quad, \quad \lim {x \rightarrow x_2^{-}} f(x) \quad, \quad \ldots \quad, \quad \lim _{x \rightarrow \beta^{-}} f(x)$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|实分析作业代写Real analysis代考|MATH351

statistics-lab™ 为您的留学生涯保驾护航 在代写实分析Real analysis方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写实分析Real analysis代写方面经验极为丰富，各种代写实分析Real analysis相关的作业也就用不着说。

## 数学代写|实分析作业代写Real analysis代考|Jordan and Hahn Decompositions

The subject of the present section is decompositions of additive and completely additive real-valued set functions into positive and negative parts. This material will be applied in Section 4 to obtain the Radon-Nikodym Theorem, an abstract generalization of some consequences of Lebesgue’s theory of differentiation of integrals. In turn, we shall use the Radon-Nikodym Theorem in Section 5 to address the subject of continuous linear functionals on $L^p$ spaces.
A real-valued additive set function $v$ on an algebra of sets is said to be bounded if $|v(E)| \leq C$ for all $E$ in the algebra. A real-valued completely additive set function on a $\sigma$-algebra of sets is said to be a signed measure.

Theorem 9.14 (Jordan decomposition). Let $v$ be a bounded additive set function on an algebra $\mathcal{A}$ of sets, and define set functions $v^{+}$and $v^{-}$on $\mathcal{A}$ by
$$v^{+}(E)=\sup {\substack{F \subseteq E \ F \in \mathcal{A}}} v(F) \text { and } \quad v^{-}(E)=-\inf {\substack{F \subseteq E, F \in \mathcal{A}}} v(F)$$

Then $v^{+}$and $v^{-}$are nonnegative bounded additive set functions on $\mathcal{A}$ such that $v=v^{+}-v^{-}$. They are completely additive if $v$ is completely additive. In any event, the decomposition $v=v^{+}-v^{-}$is minimal in the sense that an equality $\nu=\mu^{+}-\mu^{-}$in which $\mu^{+}$and $\mu^{-}$are nonnegative bounded additive set functions must have $v^{+} \leq \mu^{+}$and $v^{-} \leq \mu^{-}$.

Proof. First let us see that $v^{+}$is additive always. In fact, let $E_1$ and $E_2$ be disjoint members of $\mathcal{A}$. If $F \subseteq E_1 \cup E_2$, then the additivity of $v$ implies that $v(F)=v\left(F \cap E_1\right)+v\left(F \cap E_2\right) \leq v^{+}\left(E_1\right)+v^{+}\left(E_2\right)$. Hence
$$v^{+}\left(E_1 \cup E_2\right) \leq v^{+}\left(E_1\right)+v^{+}\left(E_2\right) .$$
On the other hand, if $F_1 \subseteq E_1$ and $F_2 \subseteq E_2$, then $v\left(F_1\right)+v\left(F_2\right)=v\left(F_1 \cup F_2\right) \leq$ $\nu^{+}\left(E_1 \cup E_2\right)$. Taking the supremum over $F_1$ and then over $F_2$ gives
$$v^{+}\left(E_1\right)+v^{+}\left(E_2\right) \leq v^{+}\left(E_1 \cup E_2\right) .$$

The function $f$ is obtained in that chapter as the derivative almost everywhere of the distribution function of $\mu$, hence as the limit of $\mu(I) / m(I)$ as intervals $I$ shrink to a point; here $m$ is Lebesgue measure. In this formulation of the result, the geometry of the line plays an essential role, and attempts to generalize to abstract settings the construction of $f$ from limits of $\mu(I) / m(I)$ have not been fruitful.

Nevertheless, the Lebesgue decomposition itself turns out to be a general measure-theory theorem, valid for any two measures in place of $\mu$ and $d x$, as long as suitable finiteness conditions are satisfied. For a reinterpretation of the results of Chapter VII, the heart of the matter is that one can tell in advance which $\mu$ ‘s have $\mu(E)=\int_E f d x$ with the singular term $\mu_s$ absent. The answer is given by the equivalent conditions of Proposition 7.11, which are taken in that chapter as a definition of “absolute continuity” of $\mu$ with respect to $d x$. The remarkable fact is that those conditions continue to be equivalent when any two finite measures replace $\mu$ and $d x$. This is the content of the Radon-Nikodym Theorem, which we shall prove in this section, and then a version of the Lebesgue decomposition will follow as a consequence.

Let $X$ be a nonempty set, and let $\mathcal{A}$ be a $\sigma$-algebra of subsets of $X$. If $\mu$ and $v$ are measures defined on $\mathcal{A}$, we say that $v$ is absolutely continuous with respect to $\mu$, written $v \ll \mu$, if $v(E)=0$ whenever $\mu(E)=0$.

# 实分析代写

## 数学代写|实分析作业代写Real analysis代考|Jordan and Hahn Decompositions

$$v^{+}(E)=\sup {\substack{F \subseteq E \ F \in \mathcal{A}}} v(F) \text { and } \quad v^{-}(E)=-\inf {\substack{F \subseteq E, F \in \mathcal{A}}} v(F)$$

$$v^{+}\left(E_1 \cup E_2\right) \leq v^{+}\left(E_1\right)+v^{+}\left(E_2\right) .$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|傅里叶分析代写Fourier analysis代考|TMA4170

statistics-lab™ 为您的留学生涯保驾护航 在代写傅里叶分析Fourier analysis方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写傅里叶分析Fourier analysis代写方面经验极为丰富，各种代写傅里叶分析Fourier analysis相关的作业也就用不着说。

## 数学代写|傅里叶分析代写Fourier analysis代考|Conditional Expectation and Dyadic Martingale Differences

We recall the definition of dyadic cubes.
Definition 5.4.1. A dyadic interval in $\mathbf{R}$ is an interval of the form
$$\left[m 2^{-k},(m+1) 2^{-k}\right)$$
where $m, k$ are integers. A dyadic cube in $\mathbf{R}^n$ is a product of dyadic intervals of the same length. That is, a dyadic cube is a set of the form
$$\prod_{j=1}^n\left[m_j 2^{-k},\left(m_j+1\right) 2^{-k}\right)$$
for some integers $m_1, \ldots, m_n, k$.
We defined dyadic intervals to be closed on the left and open on the right, so that different dyadic intervals of the same length are always disjoint sets.

Given a cube $Q$ in $\mathbf{R}^n$ we denote by $|Q|$ its Lebesgue measure and by $\ell(Q)$ its side length. We clearly have $|Q|=\ell(Q)^n$. We introduce some more notation.

Definition 5.4.2. For $k \in \mathbf{Z}$ we denote by $\mathscr{D}k$ the set of all dyadic cubes in $\mathbf{R}^n$ whose side length is $2^{-k}$. We also denote by $\mathscr{D}$ the set of all dyadic cubes in $\mathbf{R}^n$. Then we have $$\mathscr{D}=\bigcup{k \in \mathbf{Z}} \mathscr{D}_k,$$
and moreover, the $\sigma$-algebra $\sigma\left(\mathscr{D}_k\right)$ of measurable subsets of $\mathbf{R}^n$ formed by countable unions and complements of elements of $\mathscr{D}_k$ is increasing as $k$ increases.

## 数学代写|傅里叶分析代写Fourier analysis代考|Relation Between Dyadic Martingale Differences and Haar Functions

We have the following result relating the Haar functions to the dyadic martingale difference operators.

Proposition 5.4.5. For every locally integrable function $f$ on $\mathbf{R}$ and for all $k \in \mathbf{Z}$ we have the identity
$$D_k(f)=\sum_{I \in \mathscr{D}{k-1}}\left\langle f, h_I\right\rangle h_I$$ and also $$\left|D_k(f)\right|{L^2}^2=\sum_{I \in \mathscr{D}_{k-1}}\left|\left\langle f, h_I\right\rangle\right|^2 .$$

Proof. We observe that every interval $J$ in $\mathscr{D}k$ is either an $I_L$ or an $I_R$ for some unique $I \in \mathscr{D}{k-1}$. Thus we can write
\begin{aligned} & E_k(f)=\sum_{J \in \mathscr{D}k}\left(\underset{J}{\operatorname{Avg} f)} \chi_J\right. \ & =\sum{I \in \mathscr{V}{k-1}}\left[\left(\frac{2}{|I|} \int{I_L} f(t) d t\right) \chi_{I_L}+\left(\frac{2}{|I|} \int_{I_R} f(t) d t\right) \chi_{I_R}\right] . \ & \end{aligned}
But we also have
\begin{aligned} E_{k-1}(f) & =\sum_{I \in \mathscr{D}{k-1}}\left(\underset{I}{\operatorname{avg} f)} \chi_I\right. \ & =\sum{I \in \mathscr{D}{k-1}}\left(\frac{1}{|I|} \int{I_L} f(t) d t+\frac{1}{|I|} \int_{I_R} f(t) d t\right)\left(\chi_{I_L}+\chi_{I_R}\right) . \end{aligned}

# 傅里叶分析代写

## 数学代写|傅里叶分析代写Fourier analysis代考|Conditional Expectation and Dyadic Martingale Differences

5.4.1.定义$\mathbf{R}$中的二进间隔是如下形式的间隔
$$\left[m 2^{-k},(m+1) 2^{-k}\right)$$

$$\prod_{j=1}^n\left[m_j 2^{-k},\left(m_j+1\right) 2^{-k}\right)$$

5.4.2.定义对于$k \in \mathbf{Z}$，我们用$\mathscr{D}k$表示$\mathbf{R}^n$中边长为$2^{-k}$的所有并矢立方体的集合。我们也用$\mathscr{D}$表示$\mathbf{R}^n$中所有并矢立方体的集合。然后是$$\mathscr{D}=\bigcup{k \in \mathbf{Z}} \mathscr{D}_k,$$

## 数学代写|傅里叶分析代写Fourier analysis代考|Relation Between Dyadic Martingale Differences and Haar Functions

$$D_k(f)=\sum_{I \in \mathscr{D}{k-1}}\left\langle f, h_I\right\rangle h_I$$还有 $$\left|D_k(f)\right|{L^2}^2=\sum_{I \in \mathscr{D}_{k-1}}\left|\left\langle f, h_I\right\rangle\right|^2 .$$

\begin{aligned} & E_k(f)=\sum_{J \in \mathscr{D}k}\left(\underset{J}{\operatorname{Avg} f)} \chi_J\right. \ & =\sum{I \in \mathscr{V}{k-1}}\left[\left(\frac{2}{|I|} \int{I_L} f(t) d t\right) \chi_{I_L}+\left(\frac{2}{|I|} \int_{I_R} f(t) d t\right) \chi_{I_R}\right] . \ & \end{aligned}

\begin{aligned} E_{k-1}(f) & =\sum_{I \in \mathscr{D}{k-1}}\left(\underset{I}{\operatorname{avg} f)} \chi_I\right. \ & =\sum{I \in \mathscr{D}{k-1}}\left(\frac{1}{|I|} \int{I_L} f(t) d t+\frac{1}{|I|} \int_{I_R} f(t) d t\right)\left(\chi_{I_L}+\chi_{I_R}\right) . \end{aligned}

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|实分析作业代写Real analysis代考|Math444

statistics-lab™ 为您的留学生涯保驾护航 在代写实分析Real analysis方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写实分析Real analysis代写方面经验极为丰富，各种代写实分析Real analysis相关的作业也就用不着说。

## 数学代写|实分析作业代写Real analysis代考|Lebesgue Measure and Other Borel Measures

Lebesgue measure on $\mathbb{R}^1$ was constructed in Section V.1 on the ring of “elementary” sets – the finite disjoint unions of bounded intervals – and extended from there to the $\sigma$-algebra of Borel sets by the Extension Theorem (Theorem 5.5), which was proved in Section V.5. Fubini’s Theorem (Theorem 5.47) would have allowed us to build Lebesgue measure in $\mathbb{R}^N$ as an iterated product of 1-dimensional Lebesgue measure, but we postponed the construction in $\mathbb{R}^N$ until the present chapter in order to show that it can be carried out in a fashion independent of how we group 1-dimensional factors.

The Borel sets of $\mathbb{R}^1$ are, by definition, the sets in the smallest $\sigma$-algebra containing the elementary sets, and we saw readily that every set that is open or compact is a Borel set. We write $\mathcal{B}_1$ for this $\sigma$-algebra. In fact, $\mathcal{B}_1$ may be described as the smallest $\sigma$-algebra containing the open sets of $\mathbb{R}^1$ or as the smallest $\sigma$-algebra containing the compact sets. The reason that the open sets generate $\mathcal{B}_1$ is that every open interval is an open set, and every interval is a countable intersection of open intervals. Similarly the compact sets generate $\mathcal{B}_1$ because every closed bounded interval is a compact set, and every interval is the countable union of closed bounded intervals.

Now let us turn our attention to $\mathbb{R}^N$. We have already used the word “rectangle” in two different senses in connection with integration – in Chapter III to mean an $\mathrm{N}$-fold product along coordinate directions of open or closed bounded intervals, and in Chapter V to mean a product of measurable sets. For clarity let us refer to any product of bounded intervals as a geometric rectangle and to any product of measurable sets as an abstract rectangle or an abstract rectangle in the sense of Fubini’s Theorem. In $\mathbb{R}^N$, every geometric rectangle under our definition is an abstract rectangle, but not conversely.

## 数学代写|实分析作业代写Real analysis代考|Convolution

Convolution is an important operation available for functions on $\mathbb{R}^N$. On a formal level, the convolution $f * g$ of two functions $f$ and $g$ is
$$(f * g)(x)=\int_{\mathbb{R}^N} f(x-y) g(y) d y .$$

One place convolution arises is as a limit of a linear combination of translates: We shall see in Proposition 6.13 that the convolution at $x$ may be written also as $\int_{\mathbb{R}^N} f(y) g(x-y) d y$. If $f$ is fixed and if finite sets of translation operators $\tau_{y_i}$ and of weights $f\left(y_i\right)$ are given, then the value at $x$ of the linear combination $\sum_i f\left(y_i\right) \tau_{y_i}$ applied to $g$ and evaluated at $x$ is $\sum_i f\left(y_i\right) g\left(x-y_i\right)$. Corollary 6.17 will show a sense in which we can think of $\int_{\mathbb{R}^N} f(y) g(x-y) d y$ as a limit of such expressions.

To make mathematical sense out of $f * g$, let us begin with the case that $f$ and $g$ are nonnegative Borel functions on $\mathbb{R}^N$. The assertion is that $f * g$ is meaningful as a Borel function $\geq 0$. In fact, $(x, y) \mapsto f(x-y)$ is the composition of the continuous function $F: \mathbb{R}^{2 N} \rightarrow \mathbb{R}^N$ given by $F(x, y)=x-y$, followed by the Borel function $f: \mathbb{R}^N \rightarrow[0,+\infty]$. If $U$ is open in $[0,+\infty]$, then $f^{-1}(U)$ is in $\mathcal{B}N$, and Proposition 6.8 shows that $(f \circ F)^{-1}(U)=F^{-1}\left(f^{-1}(U)\right)$ is in $\mathcal{B}{2 N}$. Then the product $(x, y) \mapsto f(x-y) g(y)$ is a Borel function, and Fubini’s Theorem (Theorem 5.47) and Proposition 6.1 combine to show that $x \mapsto(f * g)(x)$ is a Borel function $\geq 0$.
Proposition 6.13. For nonnegative Borel functions on $\mathbb{R}^N$,
(a) $f * g=g * f$,
(b) $f *(g * h)=(f * g) * h$.

# 实分析代写

## 数学代写|实分析作业代写Real analysis代考|Lebesgue Measure and Other Borel Measures

$\mathbb{R}^1$的Borel集合，根据定义，是包含初等集合的最小的$\sigma$ -代数中的集合，我们很容易看到每个开或紧的集合都是Borel集合。我们把$\sigma$ -代数写成$\mathcal{B}_1$。事实上，$\mathcal{B}_1$可以被描述为包含$\mathbb{R}^1$的开集的最小的$\sigma$ -代数或包含紧集的最小的$\sigma$ -代数。开集生成$\mathcal{B}_1$的原因是每个开区间都是一个开集，每个区间都是开区间的可数交集。类似地，紧集生成$\mathcal{B}_1$，因为每个闭有界区间都是紧集，并且每个区间都是闭有界区间的可数并。

## 数学代写|实分析作业代写Real analysis代考|Convolution

$$(f * g)(x)=\int_{\mathbb{R}^N} f(x-y) g(y) d y .$$

(a) $f * g=g * f$;
(b) $f *(g * h)=(f * g) * h$。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|实分析作业代写Real analysis代考|MA50400

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## 数学代写|实分析作业代写Real analysis代考|Measurable Functions

In this section, $X$ denotes a nonempty set, and $\mathcal{A}$ is a $\sigma$-algebra of subsets of $X$. The measurable sets are the members of $\mathcal{A}$.
We say that a function $f: X \rightarrow \mathbb{R}^*$ is measurable if
(i) $f^{-1}([-\infty, c))$ is a measurable set for every real number $c$.
Equivalently the measurability of $f$ may be defined by any of the following conditions:
(ii) $f^{-1}([-\infty, c])$ is a measurable set for every real number $c$,
(iii) $f^{-1}((c,+\infty])$ is a measurable set for every real number $c$,
(iv) $f^{-1}([c,+\infty])$ is a measurable set for every real number $c$.
In fact, the implications (i) implies (ii), (ii) implies (iii), (iii) implies (iv), and (iv) implies (i) follow from the identities ${ }^4$
\begin{aligned} f^{-1}([-\infty, c]) & =\bigcap_{n=1}^{\infty} f^{-1}\left(\left[-\infty, c+\frac{1}{n}\right)\right), \ f^{-1}((c,+\infty]) & =\left(f^{-1}([-\infty,-c])\right)^c, \ f^{-1}([c,+\infty]) & =\bigcap_{n=1}^{\infty} f^{-1}\left(\left(c-\frac{1}{n},+\infty\right]\right), \ f^{-1}([-\infty, c)) & =\left(f^{-1}([-c,+\infty])\right)^c . \end{aligned}

## 数学代写|实分析作业代写Real analysis代考|Lebesgue Integral

Throughout this section, $(X, \mathcal{A}, \mu)$ denotes a measure space. The measurable sets continue to be those in $\mathcal{A}$. Our objective in this section is to define the Lebesgue integral. We defer any systematic discussion of properties of the integral to Section 4 .

Just as with the Riemann integral, the Lebesgue integral is defined by means of an approximation process. In the case of the Riemann integral, the process is to use upper sums and lower sums, which capture an approximate value of an integral by adding contributions influenced by proximity in the domain of the integrand. The process is qualitatively different for the Lebesgue integral, which captures an approximate value of an integral by adding contributions based on what happens in the image of the integrand.

Let $s$ be a simple function $\geq 0$. By our convention at the end of the previous section, we have incorporated measurability into the definition of simple function. Let $E$ be a measurable set, and let $s=\sum_{n=1}^N c_n I_{A_n}$ be the canonical expansion of $s$. We define $\mathcal{I}E(s)=\sum{n=1}^N c_n \mu\left(A_n \cap E\right)$. This kind of object will be what we use as an aproximation in the definition of the Lebesgue integral; the formula shows the sense in which $\mathcal{I}_E(s)$ is built from the image of the integrand.

If $f \geq 0$ is a measurable function and $E$ is a measurable set, we define the Lebesgue integral of $f$ on the set $E$ with respect to the measure $\mu$ to be
$$\int_E f d \mu=\int_E f(x) d \mu(x)=\sup _{\substack{0 \leq s \leq f, s \text { simple }}} \mathcal{I}_E(s) .$$

# 实分析代写

## 数学代写|实分析作业代写Real analysis代考|Measurable Functions

(i) $f^{-1}([-\infty, c))$是每个实数$c$的可测集。

(ii) $f^{-1}([-\infty, c])$是每个实数$c$的可测集;
(iii) $f^{-1}((c,+\infty])$是每个实数$c$的可测集;
(iv) $f^{-1}([c,+\infty])$是每个实数$c$的可测集。

\begin{aligned} f^{-1}([-\infty, c]) & =\bigcap_{n=1}^{\infty} f^{-1}\left(\left[-\infty, c+\frac{1}{n}\right)\right), \ f^{-1}((c,+\infty]) & =\left(f^{-1}([-\infty,-c])\right)^c, \ f^{-1}([c,+\infty]) & =\bigcap_{n=1}^{\infty} f^{-1}\left(\left(c-\frac{1}{n},+\infty\right]\right), \ f^{-1}([-\infty, c)) & =\left(f^{-1}([-c,+\infty])\right)^c . \end{aligned}

## 数学代写|实分析作业代写Real analysis代考|Lebesgue Integral

$$\int_E f d \mu=\int_E f(x) d \mu(x)=\sup _{\substack{0 \leq s \leq f, s \text { simple }}} \mathcal{I}_E(s) .$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。