# Outer Measure

• Jul 23rd 2009, 11:51 PM
Sampras
Outer Measure
So one of the axioms of outer measure is the following:

Countable subadditivity axiom: If $\displaystyle A = \bigcup_{n=1}^{\infty} A_n$ then $\displaystyle m^{*}A \leq \sum_{n=1}^{\infty} m^{*}A_n$.

Well it's not really an axiom since we want/have to prove it. But what is the point of using the "$\displaystyle \varepsilon/2^n$ trick" to prove this? E.g. we have the following:

$\displaystyle \sum_{k=1}^{\infty} |I_{kn}| < m^{*} A_n+ \frac{\varepsilon}{2^n}$ (1)

This is analogous to $\displaystyle \inf S \leq x \Rightarrow x < \inf S + \varepsilon$ for some $\displaystyle \varepsilon >0$ (with $\displaystyle x \in S$). But what is special about $\displaystyle \varepsilon/2^n$? Why not just use $\displaystyle \varepsilon$?
• Jul 24th 2009, 02:24 AM
Moo
Hello,

Because $\displaystyle \sum_{n=1}^\infty \frac{\varepsilon}{2^n}=\varepsilon$, wouldn't it ? :)
• Jul 24th 2009, 05:25 AM
Sampras
Quote:

Originally Posted by Moo
Hello,

Because $\displaystyle \sum_{n=1}^\infty \frac{\varepsilon}{2^n}=\varepsilon$, wouldn't it ? :)

yes, but the sum doesn't have to sum to $\displaystyle \varepsilon$ since it's arbitrary. I guess its convention.
• Jul 24th 2009, 06:44 AM
Moo
Quote:

Originally Posted by Sampras
yes, but the sum doesn't have to sum to $\displaystyle \varepsilon$ since it's arbitrary. I guess its convention.

Not really...

As you're dealing with a limit, you want $\displaystyle \varepsilon$ to appear. This one is arbitrary !

Since you'll sum from 1 to infinity, if you keep $\displaystyle \varepsilon$, you'll sum something positive an infinite amount of times.
Which is irrelevant for your problem.

$\displaystyle \varepsilon/2^n$ is arbitrary and > 0. It's a good candidate, because of this infinite sum.

And you'll get $\displaystyle \varepsilon$ in the end.

You could also have taken $\displaystyle \varepsilon/4^n$ or whatever you want. The main point is to get a converging series, arbitrary small.