I have just learned how to construct the Lebesgue measure. My text uses the Carathéodory's extension theorem which nowadays seems to be the most popular way to construct it (as I read here on MSE).
After going through all the definitions (Dynkin system, semi-ring, etc.) and proofs I have problems remembering the route because it seemed to come out of the blue. Just like the $\varepsilon/\delta$-proofs where things are written down in the opposite order compared to the scratch-work.
So I would like to read a coherent story about construction of Lebesgue measure (specifically, about this Carathéodory's-extension-theorem-way of constructing it). Without technicalities of course. A story written in the right order.
To my humble knowledge, the story could begin something like this:
We want to have a function which assigns some $n$-dimensional volume to each subset of $\mathbb R^n$. We have two natural wishes: the volume of the unit box should be $1$ and the function itself should have some natural properties (the definition of pre-measure here, I guess, since we don't have $\sigma$-algebra yet). Now we notice that $\mathcal P(\mathbb R^n)$ is too big for us (Vitali here), and therefore...