Constructing lebesgue measure on R

Question

I am trying to understand how these lecture notes:https://www.mat.univie.ac.at/~armin/lect/real_analysis.pdf construct lebesgue measure on page 7 and I am stuck on part 2 of theorem 2.1. Could someone please clarify the whole proof of that part? To be more specific I would like to know why can we take the diameter of each interval $Q_i$ in $R$ to be less than $\delta$ after subdividing and also how did he combine the last two infinite series into one to get the required inequality he was looking for.

Answer 1

Suppose there exists an interval with a diameter greater than $\delta$. Subdivide it by cutting it in half. Example, suppose $\delta = 0.00001$ and you have an interval $[4,5]$. You can write that as $[4,4.5]\cup [4.5,5]$, then $[4,4.25],[4.25,4.5],[4.5,4.75],[4.75,5]$, etc. And eventually, you will get every interval to have a diameter less than $\delta$. Essentially, given any $\delta>0$ and any $x \in \mathbb{R}^+$, there exists $n \in \mathbb{N}$ such that $\dfrac{x}{2^n} < \delta$. This implies a countable number of subdivisions to make sure that every interval has diameter less than $\delta$ (the union of a countable collection of countable sets is countable). Here is a proof:

Prove that the union of countably many countable sets is countable

He combined them by the same principle. If you have a countable collection of sets, then there exists a bijection between the countable collection and an index set that uses only positive even numbers. Similarly, there exists a bijection between the countable collection and an index set that uses only positive odd numbers. So, if you want to add them up together, you can sum over the index set of all positive natural numbers (odds and evens together gives all natural numbers). Basically, it is just ways to manipulate countable collections into more easily manipulable forms.