3.4 KiB
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.)