Suppose I have the disjoint sets $\{(1,2],(3,4],(0,1],(2,3],(4,5]\}$. I can reorder these in such a way that I would like to call increasing: $\{(0,1],(1,2],(2,3],(3,4],(4,5]\}$.
Similarly, suppose I have a collection of sets $\{(n,n+1]:n \in \mathbb{N}\}$. Can I reorder these in such a way to also achieve the same result.
I am asking this in relation to the answer given in Easy proof for existence of Lebesgue-premeasure to prove countable additivity of the Lebesgue pre-measure. The OP claims he can reorder disjoint intervals of the real line in such a way to produce a telescoping sum. I cannot see what is wrong with the examples given in the solution to this question, nor the approach taken by the OP.
This is the example they gave:
"Simple example: Say $I_n=[1-1/n,1-1/(n+1))$, so $[0,1)=\bigcup_{n=1}^\infty I_n$. Those intervals can be ordered the way you want, in fact they already are ordered that way.
But now say $J_n=[2-1/n,2-1/(n+1))$. So $[0,2)=\bigcup_{n=1}^ \infty I_n \cup\bigcup_{n=1}^\infty J_n$. You can't "reorder" the collection of all the $I_n$ and the $J_n$ the way you want."