An Example of Monotonicity

We continue with our example and show that the set function $\mu$ which assigns any semiclosed interval its length has various monotonicity properties.

First off, let $\{E_1,\dots,E_n\}$ be a finite, disjoint collection of semiclosed intervals, all of which are contained in another semiclosed interval $E$ . Then we have the inequality

$\displaystyle\sum\limits_{i=1}^n\mu(E_i)\leq\mu(E)$

Indeed, we can write $E=\left[a,b\right)$ , $E_i=\left[a_i,b_i\right)$ , and without loss of generality assume that $a_1<a_2<\dots<a_n$ . Then our hypotheses tell us that

$\displaystyle a\leq a_1<b_1\leq a_2<b_2\leq\dots\leq a_n<b_n\leq b$

and thus

$\displaystyle\begin{aligned}\sum\limits_{i=1}^n\mu(E_i)&=\sum\limits_{i=1}^n(b_i-a_i)\\&\leq\sum\limits_{i=1}^n(b_i-a_i)+\sum\limits_{i=1}^{n-1}(a_{i+1}-b_i)\\&=b_n-a_1\leq b-a=\mu(E)\end{aligned}$

On the other hand, if $F=\left[a,b\right]$ is a closed interval contained in the union of a finite number of bounded open intervals $U_i=\left(a_i,b_i\right)$ , then we have the strict inequality

$\displaystyle b-a<\sum\limits_{i=1}^n(b_i-a_i)$

We can rearrange the open intervals by picking $U_1$ to contain $a$ . Then if $b_1>b$ we have $F\subseteq U_1$ and we can discard all the other sets since they only increase the right hand side of the inequality. But if $b_1\leq b$ , we can pick some $U_2$ containing $b_1$ . Now we repeat, asking whether $b_2$ is greater or less than $b$ . Eventually we’ll have a finite collection of $U_i$ satisfying $a_1<a<b_1$ , $a_n<b<b_n$ , and $a_{i+1}<b_i<b_{i+1}$ . It follows that

$\displaystyle\begin{aligned}b-a&<b_n-a_1\\&=(b_1-a_1)+\sum\limits_{i=1}^{n-1}(b_{i+1}-b_i)\\&\leq\sum\limits_{i=1}^n(b_i-a_i)\end{aligned}$

What does this have to do with semiclosed intervals? Well, if $\{E_i\}_{i=1}^\infty$ is a countable sequence of semiclosed intervals that cover another semiclosed interval $E$ , then we have the inequality

$\displaystyle\mu(E)\leq\sum\limits_{i=1}^\infty\mu(E_i)$

If $E=\emptyset$ , then this is trivially true, so we’ll assume it isn’t, and let $\epsilon$ be a positive number with $\epsilon<b-a$ . Then we have the closed set $F=\left[a,b-\epsilon\right]$ . We can also pick any positive number $\delta$ and define $U_i=\left(a_i-\frac{\delta}{2^i},b_i\right)$ .

Now $F$ is smaller than $E$ , and each $U_i$ is larger than the corresponding $E_i$ , and so we find that $F$ is a closed interval covered by the open intervals $U_i$ . But the Heine-Borel theorem says that $F$ is compact, and so we can find a finite collection of the $U_i$ which cover $F$ . Renumbering the open intervals, we have

$\displaystyle F\subseteq\bigcup\limits_{i=1}^nU_i$

and our above result tells us that

$\displaystyle\begin{aligned}\mu(E)-\epsilon&=b-a-\epsilon\\&<\sum\limits_{i=1}^n\left(b_i-a_i+\frac{\delta}{2^i}\right)\\&\leq\sum\limits_{i=1}^\infty\mu(E_i)+\delta\end{aligned}$

Since we can pick $\epsilon$ and $\delta$ to be arbitrarily small, the desired inequality follows.

About these ads

April 15, 2010 - Posted by John Armstrong | Analysis, Measure Theory

1 Comment »

[...] intervals is a measure. It’s clearly real-valued and non-negative. We already showed that it’s monotonic, and this will come in handy as we show that it’s countably [...]

Pingback by An Example of a Measure « The Unapologetic Mathematician | April 16, 2010 | Reply

About this weblog

This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now.

RSS Feeds

RSS - Posts
RSS - Comments
Feedback

Got something to say? Anonymous questions, comments, and suggestions at Formspring.me!
Subjects
Archives

April 2010

M T W T F S S

« Mar May »

1 2 3 4

5 6 7 8 9 10 11

12 13 14 15 16 17 18

19 20 21 22 23 24 25

26 27 28 29 30
Search for:

The Unapologetic Mathematician

Mathematics for the interested outsider