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The general set theory claims that there exists a set which has no members. From the point of view of R what are null sets? More specifically when we define structures such as algebras or subsets of R etc.

I am reading Measure theory and in there one is talking about the algebra of sets of all finite unions of the form $(a, b]$, $(-\infty, b]$, $(a, \infty)$, $(-\infty, \infty)$ how do you show that $\emptyset$ belongs to the family F of all unions of sets as mentioned above?

One way to look at it is to say sets of measure zero are null sets, but we have not defined what lebesgue measure is yet.

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  1. The set which has no members is called the empty set. Some people mean this when they say ‘null set’, but this is confusing in the context of measure theory.

  2. The empty set is the union of no sets, just the same way that zero is the sum of no numbers.

  3. Sets of measure zero are precisely the null sets, but this is by definition. But there is another way of characterising sets of Lebesgue measure zero: a set $Y$ is a null set if and only if for all positive $\epsilon$, there exists a cover of $Y$ by countably many open intervals $U_i$ such that the sum of the lengths of all the intervals $U_i$ is less than $\epsilon$.

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Probably Zhen's answer is best, zero is finite, so finite unions include unions of zero sets. But in this case we might allow $(a,a] = \emptyset$ to get the emptyset. –  GEdgar Aug 22 '11 at 14:46
    
Thanks for the response. Using 2, I suppose the fact that the null set or empty set is union of no sets or 0 sets implies it is finite. One can claim that the $\emptyset$ therefore belongs automatically to the algebra. Using definition 3 implies that measure has already been defined. I am uncomfortable either way but I guess it is what it is. –  Ramesh Kadambi Aug 22 '11 at 15:31
    
@Ramesh: In case it wasn't clear, there are non-empty null sets. For example, $\mathbb{Q}$ is an null set in $\mathbb{R}$. –  Zhen Lin Aug 22 '11 at 15:37
    
@Zhen: I understand that there are non-empty null sets and it has to be defined according to the context. In defining the algebra one needs to already define what the measure is and the algebra seems to be interconnected. In that sense one cannot say what a null set is with out defining the measure, at least in the context of measure theory. $\mathbb{Q}$ is a null set in $\mathbb{R}$ under lebesgue measure. I think I get it. –  Ramesh Kadambi Aug 22 '11 at 16:03
    
@Ramesh: You seem to be confused about something, but I'm not sure what. $\emptyset$ is the empty set, which is a null set. The definition of an algebra of sets does not require the measure to be already defined; null sets are irrelevant to the definition. –  Zhen Lin Aug 22 '11 at 16:06
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