I am scratching my head trying to understand the notion of measurability in Fell's and Doran's book: Representations of (star)-algebras, locally compact groups, and Banach (star)-algebraic bundles.
They developed a theory of integration using $\delta$-rings instead of $\sigma$-algebras. This is supposedly an easier way of working with complex measures that ensures finite-ness.
Let $X$ be an arbitrary (nonempty) set, a $\delta$-ring $\mathscr{R}$ of $X$ is any collection of sets in $X$ that is closed with respect to finite unions, relative complements, and countable intersections.
Any finitely additive map $\mu: \mathscr{R}\to \mathbb{C}$ is called a complex measure on $\mathscr{R}.$
$A\subseteq X$ (not necessarily in $\mathscr{R})$ is said to be $\mu$-null if there exists $\{W_i\}_{i\in \mathbb{N}}\subseteq \mathscr{R}$ such that $\mu(W_i)=0$ for all $i.$
$M \subseteq X$ is said to be measurable if there exists a $B\in \mathscr{R}$ such that $M\triangle B$ (symmetric difference) is $\mu$-null.
$L\subseteq X$ is said to be locally $\mu$-measurable (locally $\mu$-null) if for all $B\in \mathscr{R}$, $L\cap B$ is $\mu$-measurable ($\mu$-null).
The following facts seem to be easy exercises:
- The collection of all $\mu$-measurable sets define a $\delta$-ring containing $\mathscr{R}$ and all $\mu$-null sets.
- The collection of all locally $\mu$-measurable sets define a $\sigma$-algebra containing all $\mu$-measurable sets and all locally $\mu$-null sets.
The claim now in question is: If $\mathscr{M}_{\mu}(\mathscr{R})$ is the collection of all $\mu$-measurable sets, then there is a unique extension of $\mu: \mathscr{R}\to\mathbb{C}$ to a complex measure $\mu': \mathscr{M}_{\mu}(\mathscr{R})\to \mathbb{C}$ such that $\mu(N)=0$ for all $\mu$-null $N$.
This claim seems to me like their version of Carathéodory extension, would anyone have an idea of a proof?
I would also greatly appreciate it if anybody knows of an analogy between the version of $\mu$-measurability above compared to the usual definition of $\mu$-measurability (via outer measure).
Edit: After checking the end-of-chapter remarks in the book, it seems that Fell and Doran has a more comprehensive reference for this flavor of measure theory. The book is: Measures and Integrals by Kelley, J. L., and Srinavasan, T. P.
Somebody in the comments below put an idea for an answer, and it is probably right. I will try to follow it and write my answer some time.
