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## 数学代写|概率论代写Probability theory代考|Partition of Unity

In this section, we define and construct a partition of unity relative to a binary approximation of a locally compact metric space $(S, d)$.

There are many different versions of partitions of unity in the mathematics literature, providing approximate linear bases in the analysis of various linear spaces of functions. The present version, roughly speaking, furnishes an approximate linear basis for $C(S, d)$, the space of continuous functions with compact supports on a locally compact metric space. In this version, the basis functions will be endowed with specific properties that make later applications simpler. For example, each basis function will be Lipschitz continuous.
First we prove an elementary lemma for Lipschitz continuous functions.
Lemma 3.3.1. Definition and basics for Lipschitz continuous functions. Let $(S, d)$ be an arbitrary metric space. A real-valued function $f$ on $S$ is said to be Lipschitz continuous, with Lipschitz constant $c \geq 0$, if $|f(x)-f(y)| \leq c d(x, y)$
for each $x, y \in S$. We will then also say that the function has Lipschitz constant $c$. Let $x_{\circ} \in S$ be an arbitrary but fixed reference point. Let $f, g$ be real-valued functions with Lipschitz constants $a, b$, respectively, on $S$. Then the following conditions hold:

1. $d\left(\cdot, x_{\circ}\right)$ has Lipschitz constant 1 .
2. $\alpha f+\beta g$ has Lipschitz constant $|\alpha| a+|\beta| b$ for each $\alpha, \beta \in R$. If, in addition, $|f| \leq 1$ and $|g| \leq 1$, then fg has Lipschitz constant $a+b$.
3. $f \vee g$ and $f \wedge g$ have Lipschitz constant $a \vee b$.
4. $1 \wedge\left(1-c d\left(\cdot, x_0\right)\right)_{+}$has Lipschitz constant $c$ for each $c>0$.
5. If $|f| \vee|g| \leq 1$, then fg has Lipschitz constant $a+b$.
6. Suppose $\left(S^{\prime}, d^{\prime}\right)$ is a locally compact metric space. Suppose $f^{\prime}$ is a realvalued function on $S^{\prime}$, with Lipschitz constant $a^{\prime}>0 .$ Suppose $|f| \vee\left|f^{\prime}\right| \leq 1$.

## 数学代写|概率论代写Probability theory代考|One-Point Compactification

The countable power of a locally compact metric space $(S, d)$ is not necessarily locally compact, while the countable power of a compact metric space remains compact. For that reason, we will often find it convenient to embed a locally compact metric space into a compact metric space such that while the metric is not preserved, the continuous functions are. This embedding is made precise in the present section, by an application of partitions of unity.

The next definition is an embellishment of definition 6.6, proposition 6.7, and theorem $6.8$ of [Bishop and Bridges 1985].

Definition 3.4.1. One-point compactification. A one-point compactification of a locally compact metric space $(S, d)$ is a compact metric space $(\bar{S}, \bar{d})$ with an element $\Delta$, called the point at infinity, such that the following five conditions hold:

1. $\bar{d} \leq 1$ and $S \cup{\Delta}$ is a dense subset of $(\bar{S}, \bar{d})$.
2. Let $K$ be an arbitrary compact subset of $(S, d)$. Then there exists $c>0$ such that $\bar{d}(x, \Delta) \geq c$ for each $x \in K$.
3. Let $K$ be an arbitrary compact subset of $(S, d)$. Let $\varepsilon>0$ be arbitrary. Then there exists $\delta_K(\varepsilon)>0$ such that for each $y \in K$ and $z \in S$ with $\bar{d}(y, z)<\delta_K(\varepsilon)$, we have $d(y, z)<\varepsilon$. In particular, the identity mapping $\bar{\imath}:(S, \bar{d}) \rightarrow(S, d)$, defined by $\bar{l}(x) \equiv x$ for each $x \in S$, is uniformly continuous on each compact subset $K$ of $(S, d)$.
4. The identity mapping $\iota:(S, d) \rightarrow(S, \bar{d})$, defined by $\iota(x) \equiv x$ for each $x \in S$, is uniformly continuous on $(S, d)$. In other words, for each $\varepsilon>0$, there exists $\delta_{\bar{d}}(\varepsilon)>0$ such that $\bar{d}(x, y)<\varepsilon$ for each $x, y \in S$ with $d(x, y)<\delta_{\bar{d}}(\varepsilon)$.
5. For each $n \geq 1$, we have
$$\left(d\left(\cdot, x_0\right)>2^{n+1}\right) \subset\left(\bar{d}(\cdot, \Delta) \leq 2^{-n}\right) .$$
Thus, as a point $x \in S$ moves away from $x_{\circ}$ relative to $d$, it converges to the point $\Delta$ at infinity relative to $\bar{d}$.
First we provide some convenient notations.

# 概率论代考

## 数学代写|概率论代写Probability theory代考|Partition of Unity

1. $d\left(\cdot, x_{\circ}\right)$ 有 Lipschitz 常数 1 。
2. $\alpha f+\beta g$ 有 Lipschitz 常数 $|\alpha| a+|\beta| b$ 对于每个 $\alpha, \beta \in R$. 如果，此外， $|f| \leq 1$ 和 $|g| \leq 1$ ，那么 fg 有 Lipschitz 常数 $a+b$.
3. $f \vee g$ 和 $f \wedge g$ 有 Lipschitz 常数 $a \vee b$.
4. $1 \wedge\left(1-c d\left(\cdot, x_0\right)\right)_{+}$有 Lipschitz 常数c对于每个 $c>0$.
5. 如果 $|f| \vee|g| \leq 1$, 那么 $\mathrm{fg}$ 有 Lipschitz 常数 $a+b$.
6. 认为 $\left(S^{\prime}, d^{\prime}\right)$ 是局部紧的度量空间。认为 $f^{\prime}$ 是一个实值函数 $S^{\prime}$, Lipschitz 常数 $a^{\prime}>0$.认为 $|f| \vee\left|f^{\prime}\right| \leq 1$.

## 数学代写|概率论代写Probability theory代考|One-Point Compactification

1. $\bar{d} \leq 1$ 和 $S \cup \Delta$ 是一个稠密的子集 $(\bar{S}, \bar{d})$.
2. 让 $K$ 是的任意紧凑子集 $(S, d)$. 那么存在 $c>0$ 这样 $\bar{d}(x, \Delta) \geq c$ 对于每个 $x \in K$.
3. 让 $K$ 是的任意紧凑子集 $(S, d)$. 让 $\varepsilon>0$ 随意。那么存在 $\delta_K(\varepsilon)>0$ 这样对于每个 $y \in K$ 和 $z \in S$ 和 $\bar{d}(y, z)<\delta_K(\varepsilon)$ ，我们有 $d(y, z)<\varepsilon$. 特别是恒等映射 $\bar{\imath}:(S, \bar{d}) \rightarrow(S, d)$ ，被定义为 $\bar{l}(x) \equiv x$ 对于每个 $x \in S$ ，在每个紧堘子集上是一致连续的 $K$ 的 $(S, d)$.
4. 身份映射 $\iota:(S, d) \rightarrow(S, \bar{d})$ ， 被定义为 $\iota(x) \equiv x$ 对于每个 $x \in S$ ，在上一致连续 $(S, d)$. 换句话说，对于每个 $\varepsilon>0$ ，那里存在 $\delta_{\bar{d}}(\varepsilon)>0$ 这样 $\bar{d}(x, y)<\varepsilon$ 对于 每个 $x, y \in S$ 和 $d(x, y)<\delta_d(\varepsilon)$.
5. 对于每个 $n \geq 1$ ，我们有
$$\left(d\left(\cdot, x_0\right)>2^{n+1}\right) \subset\left(\bar{d}(\cdot, \Delta) \leq 2^{-n}\right) .$$
因此，作为一个点 $x \in S$ 远离 $x_0$ 关系到 $d$, 收敛到一点 $\Delta$ 在无穷远处相对于 $\bar{d}$.
首先，我们提供一些方便的符号。

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