141 lines
3.4 KiB
Markdown
141 lines
3.4 KiB
Markdown
# Math 4121 Lecture 23
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## Chapter 5 Measure Theory
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### Weierstrass idea
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Define
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$$
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S_f(x) = \{(x,y)\in \mathbb{R}^2: 0\leq y\leq f(x)\}
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$$
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We take the outer content in $\mathbb{R}^2$ of $S_f(x)$ to be the area of the largest rectangle that can be inscribed in $S_f(x)$.
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$$
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(w)\int_a^b f(x) dx = c_e(S_f(x))
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$$
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We can generalize this to higher dimensions.
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#### Definition volume of rectangle
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Let $R=I_1\times I_2\times \cdots \times I_n\in \mathbb{R}^n$ be a rectangle.
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The volume of $R$ is defined as
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$$
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\text{vol}(R) = \prod_{i=1}^n \ell(I_i)
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$$
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#### Definition of outer content
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For $S\subseteq \mathbb{R}^n$, we define the outer content of $S$ as
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$$
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c_e(S) = \inf_{\{R_j\}_{j=1}^N} \sum_{j=1}^N \text{vol}(R_j)
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$$
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where $S\subseteq \bigcup_{j=1}^N R_j$ and $R_j$ are rectangles.
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Note: $\overline{\int}f(x) dx=c_e(S_f(x))$
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#### Definition of inner content
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For $S\subseteq \mathbb{R}^n$, we define the inner content of $S$ as
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$$
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c_i(S) = \sup_{\{R_j\}_{j=1}^N} \sum_{j=1}^N \text{vol}(R_j)
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$$
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where $R_j$ are disjoint rectangles $\in \mathbb{R}^n$ and $\bigcup_{j=1}^N R_j\subseteq S$.
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Note: $\underline{\int}f(x) dx=c_i(S_f(x))$
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#### Definition of Jordan measurable set
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A set $S\subseteq \mathbb{R}^n$ is said to be _Jordan measurable_ if $c_e(S)=c_i(S)$.
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and we denote the common value **content** as $c_e(S)=c_i(S)=c(S)$.
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#### Definition of interior of a set
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The interior of a set $S\subseteq \mathbb{R}^n$ is defined as
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$$
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S^\circ = \{x\in \mathbb{R}^n: B_\delta(x)\subseteq S \text{ for some } \delta > 0\}
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$$
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_It is the largest open set contained in $S$._
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#### Definition of closure of a set
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The closure of a set $S\subseteq \mathbb{R}^n$ is defined as
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$$
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\overline{S} = S\cup S'
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$$
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or equivalently,
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$$
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\overline{S} = \{x\in \mathbb{R}^n: B_\delta(x)\cap S\neq \emptyset \text{ for all } \delta > 0\}
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$$
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where $S'$ is the set of all limit points of $S$.
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_It is the smallest closed set containing $S$._
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Homework problem: Complement of the closure of $S$ is the interior of the complement of $S$, i.e.,
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$$
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(\overline{S})^c = (S^c)^\circ
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$$
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#### Definition of boundary of a set
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The boundary of a set $S\subseteq \mathbb{R}^n$ is defined as
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$$
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\partial S = \overline{S}\setminus S^\circ
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$$
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#### Proposition 5.1 (Criterion for Jordan measurability)
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Let $S\subseteq \mathbb{R}^n$ be a bounded set. Then
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$$
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c_e(S) = c_i(S)+c_e(\partial S)
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$$
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So $S$ is Jordan measurable if and only if $c_e(\partial S)=0$.
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<details>
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<summary>Proof</summary>
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Let $\epsilon > 0$, and $\{R_j\}_{j=1}^N$ be an open cover of $\partial S$. such that $\sum_{j=1}^N \text{vol}(R_j) < c_e(\partial S)+\frac{\epsilon}{2}$.
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We slightly enlarge each $R_j$ to $Q_j$ such that $R_j\subseteq Q_j$ and $\text{vol}(Q_j)\leq \text{vol}(R_j)+\frac{\epsilon}{2N}$.
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and $dis(R_j,Q_j^c)>\delta > 0$
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If we could construct such $\{Q_j\}_{j=N+1}^M$ disjoint and
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$$
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\bigcup_{j=N+1}^M Q_j\subseteq S\subseteq \bigcup_{j=1}^M Q_j
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$$
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then we have
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$$
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c_e(S)\leq \sum_{j=1}^M \text{vol}(\partial S)+\epsilon +c_i(S)
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$$
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We can do this by constructing a set of square with side length $\eta$. We claim:
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If $\eta$ is small enough (depends on $\delta$), then $\mathcal{C}_\eta=\{Q\in K_\eta:Q\subset S\}$, $\mathcal{C}_\eta\cup \left(\bigcup_{j=1}^N Q_j\right)$ is a cover of $S$.
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Suppose $\exists x\in S$ but not in $\mathcal{C}_\eta$. Then $x$ is closed to $\partial S$ so in some $Q_j$. (This proof is not rigorous, but you get the idea. Also not clear in book actually.)
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</details>
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