David Williams constructs the Lebesgue measure $\mu$ on $\mathbb{R}$ (excerpt at the end of the post) by first defining $\mu$ on an algebra $\Sigma_0$, and then using Carathéodory's Extension Theorem (excerpt also at the end) to extend $\mu$ into $\mathcal{B}(\mathbb{R})=\{\text{Borel sets}\}$. The resulting function is not defined on $\mathcal{L}(\mathbb{R})=\{\text{Carathéodory measurable sets}\}$, but rather it is defined on a proper subset of it.
Can we define the Lebesgue measure $\mu:\mathcal{L}(\mathbb{R})\to[0,\infty]$ using Carathéodory's Extension Theorem? The goal, of course, if to define the Lebesgue measure (with $\mathcal{L}(\mathbb{R})$ as its domain) while following Williams' outline of its construction.
This could be done, for example, by finding a suitable algebra $\Sigma_0$ such that $\sigma(\Sigma_0)=\mathcal{L}(\mathbb{R})$.
