Countable Sets and Measure ZeroDate: 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. |
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