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 July 25th, 2013, 03:37 AM #1 Senior Member   Joined: Feb 2013 Posts: 153 Thanks: 0 Measure 1. What is Lebesgue measure of set N? (I found somewhere it's infinity, but, i think it's zero...) 2. If i have infinite sum of countable many functions (sum of some sequence), can I write this as countable many infinite sums sums of that function? For example, SUM(a1+a2+....)=SUM(a1)+SUM(a2)+... (where sum is infinite, for example from 0 to infinity).
July 25th, 2013, 12:36 PM   #2
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Re: Measure

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 Originally Posted by limes5 1. What is Lebesgue measure of set N? (I found somewhere it's infinity, but, i think it's zero...) 2. If i have infinite sum of countable many functions (sum of some sequence), can I write this as countable many infinite sums sums of that function? For example, SUM(a1+a2+....)=SUM(a1)+SUM(a2)+... (where sum is infinite, for example from 0 to infinity).
1) On the real line, the Lebesgue measure of any countable set is 0. N (integers) is a countable set.
2) If the sum is absolutely convergent, yes. Otherwise - maybe.

 July 26th, 2013, 04:12 AM #3 Senior Member   Joined: Feb 2013 Posts: 153 Thanks: 0 Re: Measure For the first question. So, measure of countable set is 0. If measure is 0, is that set countabe? Is that If and only if?
July 26th, 2013, 12:28 PM   #4
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Re: Measure

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 Originally Posted by limes5 For the first question. So, measure of countable set is 0. If measure is 0, is that set countabe? Is that If and only if?
No. It is possible (with a lttle work) to construct non-countable sets of measure 0.

July 26th, 2013, 12:47 PM   #5
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Re: Measure

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 Originally Posted by limes5 For the first question. So, measure of countable set is 0. If measure is 0, is that set countabe? Is that If and only if?
The Cantor set is the classic example of an uncountable set of measure zero.

https://en.wikipedia.org/wiki/Cantor_set

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