Countable Sets and Measure Zero

Date: 05/12/2001 at 19:47:18
From: Jon Klassen
Subject: Countable sets and measure zero

How would you prove that if a set S is countable, then S has measure 
zero?

Date: 05/13/2001 at 08:18:54
From: Doctor Paul
Subject: Re: Countable sets and measure zero

if S is countable, then we can write down the elements of S:

     S = {s1, s2, s3, ...}

Now recall what it means for a set to have measure zero. It means that 
given any epsilon > 0, we can cover S with a countable number of 
intervals, rectangles, cubes, or "boxes" (depending on whether we're 
talking about R^1, R^2, R^3, or R^k for k > 3) that satisfies this 
property:

   the sum of the "volumes" (use length or area if appropriate) of 
   these "boxes" is less than epsilon.

Since we're in R^1, we need to cover S with a countable number of 
intervals such that the sum of the lengths of the intervals is less 
than epsilon.

It's not an obvious proof, but once you see it, it's easy.

I'm going to use e for epsilon.

Put a disk around s1 of radius e/4. So you have essentially put a disk 
that covers the interval {s1 - e/8 , s1 + e/8).

Now go to s2. Put a disk around s2 of radius e/8: you get
(s2 - e/16 , s2 + e/16)

Do you see the pattern?

When it's all said and done, you have an infinite number of intervals 
whose lengths are:

     e/4, e/8, e/16, e/32, ...

We want to add these up and show that the total is less than epsilon.

By summing an infinite geometric series, we see that the sum of the 
lengths of these intevals is e/2 < e, and that completes the proof.

This shows, for instance, that the rationals in the intervals [0,1] 
has measure zero.

A good follow-up question would be to ask whether or not the entire 
interval [0,1] has measure zero, so what I'm asking is this: if we 
took the rationals in the interval [0,1] (which is certainly a 
countable set) and covered them using the technique described above, 
would we have covered the entire interval [0,1]? If we had covered the 
entire interval [0,1] that would mean that the interval [0,1] has 
measure zero as well.

But the interval [0,1] does not have measure zero and so it is clear 
that covering the rationals from in the interval [0,1] using this (or 
any other) method would in fact leave an uncountable number of 
irrationals uncovered.

- Doctor Paul, The Math Forum
  http://mathforum.org/dr.math/   

Date: 05/14/2001 at 20:14:19
From: Jon Klassen
Subject: Re: Countable sets and measure zero

Thank you very much, Dr. Paul.