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## 数学代写|概率论代写Probability theory代考|Riemann–Stieljes Integral

Definition 4.1.1. Distribution function. A distribution function is a nondecreasing real-valued function $F$ on $R$, with $\operatorname{domain}(F)$ dense in $R$.

Definition 4.1.2. Riemann-Stieljes sum relative to a distribution function and a partition of the real line $R$. Let $F$ be a distribution function, and let $X \in C(R)$ be arbitrary. An arbitrary finite and increasing sequence $\left(x_0, \ldots, x_n\right)$ in domain $F)$ is called a partition of $R$ relative to the distribution function $F$. One partition is said to be a refinement of another if the former contains the latter as a subsequence.

For any partition $\left(x_1, \ldots, x_n\right)$ relative to the distribution function $F$, define its mesh as
$$\operatorname{mesh}\left(x_1, \ldots, x_n\right) \equiv\left(\bigvee_{i=1}^n\left(x_i-x_{i-1}\right)\right)$$
and define the Riemann-Stieljes sum as
$$S\left(x_0, \ldots, x_n\right) \equiv \sum_{i=1}^n X\left(x_i\right)\left(F\left(x_i\right)-F\left(x_{i-1}\right)\right)$$

Theorem 4.1.3. Existence of Riemann-Stieljes integral. Let $F$ be a distribution function, and let $X \in C(R)$ be arbitrary. Then the Riemann-Stieljes sum converges as $\operatorname{mesh}\left(x_1, \ldots, x_n\right) \rightarrow 0$ with $x_0 \rightarrow-\infty$ and $x_n \rightarrow+\infty$. The limit will be called the Riemann-Stieljes integral of $X$ with respect to the distribution function $F$, and will be denoted by $\int_{-\infty}^{+\infty} X(x) d F(x)$, or simply by $\int X(x) d F(x)$.
Proof. 1. Suppose $X$ has modulus of continuity $\delta_X$ and vanishes outside the compact interval $[a, b]$, where $a, b \in \operatorname{domain}(F)$. Let $\varepsilon>0$ be arbitrary. Consider an arbitrary partition $\left(x_0, \ldots, x_n\right)$ with (i) $x_0<a-2<b+2<x_n$ and (ii) $\operatorname{mesh}\left(x_1, \ldots, x_n\right)<1 \wedge \delta_X(\varepsilon)$, where $\delta_X$ is a modulus of continuity for $X$.

## 数学代写|概率论代写Probability theory代考|Integration on a Locally Compact Metric Space

In this section, the Riemann-Stieljes integration is generalized to functions $X \in$ $C(S, d)$, where $(S, d)$ is a locally compact metric space $(S, d)$.

Traditionally, integration is usually defined in terms of a measure, a function on a family of subsets that is closed relative to the operations of countable unions, countable intersections, and relative to complements. In the case of a metric space, one such family can be generated via these three operations from the family of all open subsets. Members of the family thus generated are called Borel sets. In the special case of $R$, the open sets can in turn be generated from a countable subfamily of intervals in successive partitions of $R$, wherein ever smaller intervals cover any compact interval in $R$. The intervals in the countable family can thus serve as building blocks in the analysis of measures on $R$.

The Daniell integration theory is a more natural choice for the constructive development. Integrals of functions, rather than measures of sets, are the starting point. In the special case of a locally compact metric space $(S, d)$, the family $C(S, d)$ of continuous functions with compact supports supplies the basic integrable functions.

Definition 4.2.1. Integration on a locally compact metric space. An integration on a locally compact metric space $(S, d)$ is a real-valued linear function $I$ on the linear space $C(S, d)$ such that (i) $I(X)>0$ for some $X \in C(S, d)$ and (ii) for each $X \in C(S, d)$ with $I(X)>0$, there exists a point $x$ in $S$ for which $X(x)>0$. Condition (ii) will be called the positivity condition of the integration $I$.

It immediately follows that if $X \in C(S, d)$ is such that $X \leq 0$, then $I(X) \leq 0$. By the linearity and the positivity of the function $I$, we see that if $X \in C(S, d)$ is such that $X \geq 0$, then $I(X) \geq 0$.

# 概率论代考

## 数学代写|概率论代写Probability theory代考|Riemann–Stieljes Integral

$$\operatorname{mesh}\left(x_1, \ldots, x_n\right) \equiv\left(\bigvee_{i=1}^n\left(x_i-x_{i-1}\right)\right)$$

$$S\left(x_0, \ldots, x_n\right) \equiv \sum_{i=1}^n X\left(x_i\right)\left(F\left(x_i\right)-F\left(x_{i-1}\right)\right)$$

## 数学代写|概率论代写Probability theory代考|Integration on a Locally Compact Metric Space

Daniell整合理论是建设性发展的更自然的选择。函数的积分，而不是集合的度量，是起点。在同部紧致度量空间的特殊情况下 $(S, d)$ ，家庭 $C(S, d)$ 具有紧凑支持 的连续函数提供了基本的可积函数。

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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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