I am studying for an exam and I am trying to work out a concise construction of the Lebesgue measure. Here is what I have:
If $\mathcal{A}$ is an algebra of open intervals in $\mathbb{R}$ with a premeasure $\mu_0:\mathcal{A}\to[0,\infty)$ where $\mu_0(A)$ is the length of $A\in\mathcal{A}$. Now we want to define the outer measure $\mu^*:\mathcal{P}(\mathbb{R})\to[0,\infty]$ such that if $E\subset\mathbb{R}$, take $E\subset\bigcup_{i=1}^\infty A_i$ for $A_i\in\mathcal{A}$ for all $i$ such that $\mu^*(E)=\inf\sum_i\mu_0(A_i)$. We can show then that $\mu^*(\emptyset)=0$ and $\sigma$-additivity on $\mu^*$. Now, by Caratheodory, we have that $\mu^*$ is a complete measure in the $\sigma$-algebra of $\mu^*$ measurable sets, $\mathcal{M}$ (i.e. a set $K$ is $\mu^*$ measurable if $\mu^*(J)=\mu^*(J\cap K)+\mu^*(J\cap K^C)$ for all $J\subset\mathbb{R}$). But then $\mathcal{A}\subset\mathcal{M}$ which means $\mu^*$ is the unique completion of $\mu_0$, so $\mathcal{M}=\sigma(\mathcal{A})=\mathcal{B}_\mathbb{R}$. So, to give us what we want, we have a complete measure on $\mathcal{B}_\mathbb{R}$ that defines the length of intervals. This means the measure is the Lebesgue measure. And there we have it.
The biggest questions I have about this:
1. I know I didn't define $\mathcal{A}$ very specifically, is that ok?
2. I am fairly confident in the beginning of the construction, I am a little more unsure on how to get the complete measure that I have established from Caratheodory to be the Lebesgue measure.
3. I didn't use $F(x)=x$, which really defines the Lebesgue measure. I think I just found a way around that, but is it not ok to leave that out?