the outer measure of an interval is its length

Proof

Suppose $I = [a, b]$ . Then $I \subset (a - ϵ, b + ϵ)$ for all $ϵ > 0$ . But then we have

m^{*} (I) \leq ℓ (a - ϵ, b + ϵ) = b - a + 2 ϵ

Thus we have $m^{*} (I) \leq b - a$ .

Now we show $b - a \leq m^{*} (I)$ . Let ${I_{n}}_{n}$ be a collection of open intervals such that $[a, b] \subset ⋃_{n} I_{n}$ .

Now, since $[a, b]$ is compact (by the Heine-Borel theorem), there exists a finite subcover ${J_{k}}_{k = 0}^{m} \subset {I_{n}}_{n}$ such that $[a, b] \subset ⋃_{k = 0}^{m} J_{k}$

Since $a \in ⋃_{k = 0}^{m} J_{k}$ , there exists $k_{1}$ such that $a \in J_{k_{1}}$ . Rearranging intervals, I let $k_{1} = 1$ and $a \in J_{1} = (a_{1}, b_{1})$ . If $b_{1} < b$ , then $b_{1} \in [a, b]$ . So ther eis some $k_{2}$ such that $b_{1} \in J_{k_{2}}$ . By rearranging again, I assume $k_{2} = 2$ . So $b_{1} \in J_{2} = (a_{2}, b_{2})$ .

Continuing in the same manner, we conclude that there exists a $K, 1 \leq K \leq m$ such that $b < b_{K}$ . And for all $k \in {1, 2, \dots, K - 1}$ we have $b_{k} \leq b$ and $a_{k + 1} \leq b_{k} < b_{k + 1}$ .

But then we have

\begin{aligned} \sum_{n} ℓ (I_{n}) & \geq \sum_{k = 1}^{m} ℓ (J_{k}) \\ \geq \sum_{k = 1}^{K} ℓ (J_{k}) \\ = (b_{K} - a_{K}) + (b_{K - 1} - a_{K - 1}) + \dots + (b_{1} - a_{1}) \\ = b_{K} + (b_{K - 1} - a_{K}) + (b_{K - 2} - a_{K - 1}) + \dots + (b_{1} - a_{2}) - a_{1} \\ \geq b_{K} - a_{1} \\ \geq b - a \\ = ℓ (I) \end{aligned}

Thus we have $m^{*} (I) \geq ℓ (I) = b - a$

Thus we conclude $m^{*} (I) = ℓ (I)$

\begin{matrix} ◼ \end{matrix}

Mentions

const modules = await cJS()

const COLUMNS = [  
	{ id: "Name", value: page => page.$link },  
	{ id: "Last Modified", value: page => modules.dateTime.getLastMod(page) },
];  
  
return function View() {  
	const current = dc.useCurrentFile();
// Selecting `#game` pages, for example. 
	let queryString = `@page and linksto(${current.$link})`;
	let pages = dc.useQuery(queryString);
	
	// check types
	pages = pages.filter( (p) => !modules.typeCheck.checkAll(p, current) ).sort()
	
	
	return <dc.Table columns={COLUMNS} rows={pages} paging={20}/>;  
}

the outer measure of an interval is its length

References

See Also

Mentions