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 $c_e(\partial S)=0$ .

Proof

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.)