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## 数学代写|几何测度论代写geometric measure theory代考|Lusin’s theorem and density of continuous functions

Integration with respect to a Radon measure $\mu$ on $L^1\left(\mathbb{R}^n, \mu\right)$ is characterized by its behavior on $C_{\mathrm{c}}^0\left(\mathbb{R}^n\right)$ as a consequence of Lusin’s theorem.

Theorem 4:1 (Lusin’s theorem) If $\mu$ is a Borel measure on $\mathbb{R}^n, u: \mathbb{R}^n \rightarrow \mathbb{R}$ is a Borel function, and $E \subset \mathbb{R}^n$ a Borel set with $\mu(E)<\infty$, then for every $\varepsilon>0$ there exists a compact set $K \subset E$ such that $u$ is continuous on $K$ and
$$\mu(E \backslash K)<\varepsilon .$$ Remark 4.2 If $E \subset \mathbb{R}^n$ is a Borel set in $\mathbb{R}^n$, then $u: E \rightarrow[-\infty, \infty]$ is a Borel function provided each ${x \in E: u(x)>t}(t \in \mathbb{R})$ is a Borel set. If $\mathbb{R}^n \backslash E$ is non-empty, then, setting $u=0$ on $\mathbb{R}^n \backslash E$, we extend $u$ to the whole space $\mathbb{R}^n$ as a Borel function. Borel functions usually arise with a domain of definition properly contained in $\mathbb{R}^n$, and are tacitly extended to $\mathbb{R}^n$ as explained.

Proof of Theorem 4.1 For $k \in \mathbb{N}$, let $\left{I_h^k\right}_{h \in \mathbb{N}}$ be a countable partition of $u\left(\mathbb{R}^n\right)$ by (non-empty) Borel sets with $\operatorname{diam}\left(I_h^k\right) \leq 1 / k$, and consider the countable partition of $E$ by Borel sets $\left{E_h^k=E \cap u^{-1}\left(I_h^k\right)\right}_{h \in \mathbb{N}}$. For $k, h \in \mathbb{N}$, we choose $y_h^k \in u\left(E_h^k\right)$, and apply Theorem $2.8$ to find a compact set $K_h^k$ with
$$K_h^k \subset E_h^k, \quad \mu\left(E_h^k \backslash K_h^k\right) \leq \frac{\varepsilon}{2^{h+k}} .$$

## 数学代写|几何测度论代写geometric measure theory代考|Riesz’s theorem and vector-valued Radon measures

If $\mu$ is a Radon measure on $\mathbb{R}^n$, then the linear functional $L: C_{\mathrm{c}}^0\left(\mathbb{R}^n\right) \rightarrow \mathbb{R}$,
$$\langle L, \varphi\rangle=\int_{\mathbb{R}^n} \varphi \mathrm{d} \mu, \quad \varphi \in C_{\mathrm{c}}^0\left(\mathbb{R}^n\right),$$
is positive $\left(\varphi \geq 0\right.$ implies $\langle L, \varphi\rangle \geq 0$ ) or, equivalently, monotone $\left(\varphi_1 \leq \varphi_2\right.$ implies $\left.\left\langle L, \varphi_1\right\rangle \leq\left\langle L, \varphi_2\right\rangle\right)$. As a consequence, $L$ is continuous with respect to the following notion of convergence on $C_{\mathrm{c}}^0\left(\mathbb{R}^n\right): \varphi_h \rightarrow \varphi$ in $C_{\mathrm{c}}^0\left(\mathbb{R}^n\right)$ if $\varphi_h \rightarrow \varphi$ uniformly on $\mathbb{R}^n$ and, for a compact set $K \subset \mathbb{R}^n$,
$$\operatorname{spt} \varphi \cup \bigcup_{h \in \mathbb{N}} \operatorname{spt} \varphi_h \subset K .$$
Indeed, $\varphi_h \rightarrow \varphi$ in $C_c^0\left(\mathbb{R}^n\right)$ implies $\left\langle L, \varphi_h\right\rangle \rightarrow\langle L, \varphi\rangle$, as we clearly have
$$\sup \left{\langle L, \varphi\rangle: \varphi \in C_{\mathrm{c}}^0\left(\mathbb{R}^n\right),|\varphi| \leq M, \operatorname{spt} \varphi \subset K\right} \leq M \mu(K)<\infty,$$ for every compact set $K \subset \mathbb{R}^n$ and $M>0$. We are now going to prove that Radon measures can actually be characterized as bounded monotone linear functionals on $C_{\mathrm{c}}^0\left(\mathbb{R}^n\right)$, and use this point of view to introduce the important notion of vector-valued Radon measure. Indeed, let us consider a linear functional $L: C_{\mathrm{c}}^0\left(\mathbb{R}^n: \mathbb{R}^m\right) \rightarrow \mathbb{R}$. By linearity. $L$ is continuous with respect to convergence in $C_{\mathrm{c}}^0\left(\mathbb{R}^n ; \mathbb{R}^m\right)$ if and only if it is bounded (see Exercise 4.15), in the sense that, for every compact set $K \subset \mathbb{R}^n$,
$$\sup \left{\langle L, \varphi\rangle: \varphi \in C_{\mathrm{c}}^0\left(\mathbb{R}^n ; \mathbb{R}^m\right), \operatorname{spt} \varphi \subset K,|\varphi| \leq 1\right}<\infty .$$
If $L$ is integration with respect to a Radon measure $\mu$ on $\mathbb{R}^n$, then $L$ is a linear bounded functional on $C_{\mathrm{c}}^0\left(\mathbb{R}^n\right)$, with the additional property of being monotone. We may construct further examples.

# 几何测度论代考

## 数学代写|几何测度论代写geometric measure theory代考|Lusin’s theorem and density of continuous functions

$$\mu(E \backslash K)<\varepsilon .$$ 备注 4.2 如果 $E \subset \mathbb{R}^n$ 是一个 Borel 设置在 $\mathbb{R}^n$ ，然后 $u: E \rightarrow[-\infty, \infty]$ 是每个提供的 Borel 函数 $x \in E: u(x)>t(t \in \mathbb{R})$ 是一个 Borel 集。如果趶 $\backslash E$ 是非空

$$K_h^k \subset E_h^k, \quad \mu\left(E_h^k \backslash K_h^k\right) \leq \frac{\varepsilon}{2^{h+k}}$$

## 数学代写|几何测度论代写geometric measure theory代考|Riesz’s theorem and vector-valued Radon measures

$$\langle L, \varphi\rangle=\int_{\mathbb{R}^n} \varphi \mathrm{d} \mu, \quad \varphi \in C_{\mathrm{c}}^0\left(\mathbb{R}^n\right),$$

$$\operatorname{spt} \varphi \cup \bigcup_{h \in \mathbb{N}} \operatorname{spt} \varphi_h \subset K .$$

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assignmentutor™您的专属作业导师
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