# Math 4121 Lecture 23

## Chapter 5 Measure Theory

### Weierstrass idea

Define

$$
S_f(x) = \{(x,y)\in \mathbb{R}^2: 0\leq y\leq f(x)\}
$$

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 $S_f(x)$.

$$
(w)\int_a^b f(x) dx = c_e(S_f(x))
$$

We can generalize this to higher dimensions.

#### Definition volume of rectangle

Let $R=I_1\times I_2\times \cdots \times I_n\in \mathbb{R}^n$ be a rectangle.

The volume of $R$ is defined as

$$
\text{vol}(R) = \prod_{i=1}^n \ell(I_i)
$$

#### Definition of outer content

For $S\subseteq \mathbb{R}^n$, we define the outer content of $S$ as

$$
c_e(S) = \inf_{\{R_j\}_{j=1}^N} \sum_{j=1}^N \text{vol}(R_j)
$$

where $S\subseteq \bigcup_{j=1}^N R_j$ and $R_j$ are rectangles.

Note: $\overline{\int}f(x) dx=c_e(S_f(x))$

#### Definition of inner content

For $S\subseteq \mathbb{R}^n$, we define the inner content of $S$ as

$$
c_i(S) = \sup_{\{R_j\}_{j=1}^N} \sum_{j=1}^N \text{vol}(R_j)
$$

where $R_j$ are disjoint rectangles $\in \mathbb{R}^n$ and $\bigcup_{j=1}^N R_j\subseteq S$.

Note: $\underline{\int}f(x) dx=c_i(S_f(x))$

#### Definition of Jordan measurable set

A set $S\subseteq \mathbb{R}^n$ is said to be _Jordan measurable_ if $c_e(S)=c_i(S)$.

and we denote the common value **content** as $c_e(S)=c_i(S)=c(S)$.

#### Definition of interior of a set

The interior of a set $S\subseteq \mathbb{R}^n$ is defined as

$$
S^\circ = \{x\in \mathbb{R}^n: B_\delta(x)\subseteq S \text{ for some } \delta > 0\}
$$

_It is the largest open set contained in $S$._

#### Definition of closure of a set

The closure of a set $S\subseteq \mathbb{R}^n$ is defined as

$$
\overline{S} = S\cup S'
$$

or equivalently,

$$
\overline{S} = \{x\in \mathbb{R}^n: B_\delta(x)\cap S\neq \emptyset \text{ for all } \delta > 0\}
$$

where $S'$ is the set of all limit points of $S$.

_It is the smallest closed set containing $S$._

Homework problem: Complement of the closure of $S$ is the interior of the complement of $S$, i.e.,

$$
(\overline{S})^c = (S^c)^\circ
$$

#### Definition of boundary of a set

The boundary of a set $S\subseteq \mathbb{R}^n$ is defined as

$$
\partial S = \overline{S}\setminus S^\circ
$$

#### Proposition 5.1 (Criterion for Jordan measurability)

Let $S\subseteq \mathbb{R}^n$ be a bounded set. Then

$$
c_e(S) = c_i(S)+c_e(\partial S)
$$

So $S$ is Jordan measurable if and only if $c_e(\partial S)=0$.

<details>
<summary>Proof</summary>

Let $\epsilon > 0$, and $\{R_j\}_{j=1}^N$ be an open cover of $\partial S$. such that $\sum_{j=1}^N \text{vol}(R_j) < c_e(\partial S)+\frac{\epsilon}{2}$.

We slightly enlarge each $R_j$ to $Q_j$ such that $R_j\subseteq Q_j$ and $\text{vol}(Q_j)\leq \text{vol}(R_j)+\frac{\epsilon}{2N}$.

and $dis(R_j,Q_j^c)>\delta > 0$

If we could construct such $\{Q_j\}_{j=N+1}^M$ disjoint and

$$
\bigcup_{j=N+1}^M Q_j\subseteq S\subseteq \bigcup_{j=1}^M Q_j
$$

then we have

$$
c_e(S)\leq \sum_{j=1}^M \text{vol}(\partial S)+\epsilon +c_i(S)
$$

We can do this by constructing a set of square with side length $\eta$. We claim:

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 $S$.

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.)

</details>