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Prove that every nonempty open subset of $\mathbb{R}$ can be expressed as a countable disjoint union of open intervals:

$G=\bigcup_{k}(a_k,b_k,),$

where the range on $k$ can be finite or infinite. Furthermore, show that this expression is unique except for the numbering of the component intervals. (Hint: for any $x\in G$, show that there exists a largest open interval $A_x$ such that $x\in A_x$ and $A_x \subset G.$)

Can you give me the proof using above hint? and another problem is:

In the notation of above problem, prove that $\lambda(G)=\sum_{k}(b_k-a_k).$

SW PARK
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  • Welcome to Math.SE. Your question has already been asked and answered here; see my previous comment. Also, this site doesn't generally accept questions that just say "give me the proof"; you are expected to show some effort of your own. See http://meta.math.stackexchange.com/questions/1803/how-to-ask-a-homework-question – Nate Eldredge Mar 31 '15 at 14:03
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    Your second problem follows immediately from the first using the countable additivity of $\lambda$. – Nate Eldredge Mar 31 '15 at 14:03

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