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I am reading Sheldon Axler's book: Measure, Integration & Real Analysis ... and I am focused on Chapter 2: Measures ...

I need help with the proof of Result 2.8 ...

Result 2.8 and its proof read as follows

In the above text from Axler we read:"The doubly indexed collection of open intervals \(\displaystyle \{ I_{ j, k } : j,k \in \mathbb{Z^+} \} \) can be rearranged into a sequence of open intervals whose union contains \(\displaystyle \bigcup_{ k = 1 }^{ \infty} A_k \) as follows, where in step k (start with k =2, then k = 3,4,5 ... ) we adjoin the k-1 intervals whose indices add up to k :
Inequality 2.9 shows that the sum of the lengths listed above is less than or equal to \(\displaystyle \epsilon + \sum_{k=1}^{ \infty} |A_k| \). Thus \(\displaystyle | \sum_{k=1}^{ \infty} A_k | \leq \epsilon + \sum_{k=1}^{ \infty} |A_k| \) ... ..."

i really do not understand what is going on here ... can someone explain why we are arranging or grouping the intervals \(\displaystyle \{ I_{ j, k } : j,k \in \mathbb{Z^+} \} \) in this way and why exactly it follows that the sum of the lengths listed above is less than or equal to $ \epsilon + \sum_{k=1}^{ \infty} |A_k| $ ...Hope someone can help,

Peter

NOTE: so that readers of the above post will have enough contextual and notational information i am posting the start of Axler's Section @A on Outer Measure ... as follows:

Hope that helps Peter

I need help with the proof of Result 2.8 ...

Result 2.8 and its proof read as follows

In the above text from Axler we read:"The doubly indexed collection of open intervals \(\displaystyle \{ I_{ j, k } : j,k \in \mathbb{Z^+} \} \) can be rearranged into a sequence of open intervals whose union contains \(\displaystyle \bigcup_{ k = 1 }^{ \infty} A_k \) as follows, where in step k (start with k =2, then k = 3,4,5 ... ) we adjoin the k-1 intervals whose indices add up to k :

i really do not understand what is going on here ... can someone explain why we are arranging or grouping the intervals \(\displaystyle \{ I_{ j, k } : j,k \in \mathbb{Z^+} \} \) in this way and why exactly it follows that the sum of the lengths listed above is less than or equal to $ \epsilon + \sum_{k=1}^{ \infty} |A_k| $ ...Hope someone can help,

Peter

NOTE: so that readers of the above post will have enough contextual and notational information i am posting the start of Axler's Section @A on Outer Measure ... as follows:

Hope that helps Peter

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