Extension of measures in the real plane

Question

Can a non negative function $\mu$ defined in the Borel rectangles on the real plane be extended to a measure? Paul Halmos claims so in his book Introduction to Hilbert Space and the Theory of Spectral Multiplicity [proof of the spectral theorem for normal operators]. Refining my question, the Borel rectangles are an elementary family, and the $\sigma$-algebra generated by them is the $\sigma$-algebra of Borel in the real plane, how do you extend this function to a measure in this $\sigma$-algebra?

Answer 1

Given a countably additive function $\rho$ on an elementary family $\mathcal{E}$ such that $\rho(\emptyset)=0$, there exists a unique premeasure $\mu_0$ defined on the algebra $\mathcal{A}(\mathcal{E})$ generated by $\mathcal{E}$. Then define an outer measure $\mu^*$ via $$ \mu^*(A) := \inf\left\{\sum_1^\infty\mu_0(E_j) : E_j\in\mathcal{A}(\mathcal{E})\ \text{and}\ A \subseteq \bigcup_1^\infty E_j\right\} $$ and apply Carathéodory's theorem to get a measure $\mu$ whose measurable sets include $\sigma(\mathcal{E})$ and $\mu(E)=\mu^*(E)=\mu_0(E)=\rho(E)$ for every $E\in\mathcal{E}$.

Then you just have to see that $\sigma(\mathcal{E})$ is the collection of Borel sets in the plane. This follows from the fact that every open set is in $\sigma(\mathcal{E})$, as every open set is a countable union of open rectangles.