Let
- $\Omega$ be a set
- $\mathcal S\subseteq2^\Omega$ be a set with $\emptyset\in\mathcal S$
- $\mathcal S_{\text{loc}}:=\left\{A\subseteq\Omega:A\cap S\in\mathcal S\text{ for all }S\in\mathcal S\right\}$
- $E$ be a normed $\mathbb R$-vector space
- $\mu:\mathcal S\to E$ with $\mu(\emptyset)=0$
Now, let $$\operatorname{Var}_\mu^{\mathcal S}(A):=\sup\left\{\sum_{i=1}^k\left\|\mu(R_i)\right\|_E:k\in\mathbb N\text{ and }R_1,\ldots,R_k\in\mathcal S\text{ are pairwise disjoint with }\biguplus_{i=1}^kR_i\subseteq A\right\}$$ for $A\subseteq\Omega$.
It's easy to see that $\operatorname{Var}_\mu^{\mathcal S}$ is monotone. Moreover, if $I$ is a set and $(A_i)_{i\in I}\subseteq2^\Omega$ is disjoint, then $$\operatorname{Var}_\mu^{\mathcal S}\left(\biguplus_{i\in I}A_i\right)\ge\sum_{i\in I}\operatorname{Var}_\mu^{\mathcal S}(A_i)\;.\tag1$$ If $(A_i)_{i\in I}\subseteq\mathcal S_{\text{loc}}$ and $(\mu(S_i))_{i\in I}$ is summable with $$\sum_{i\in I}\mu(S_i)=\mu\left(\biguplus_{i\in I}S_i\right)\tag2$$ for all disjoint $(S_i)_{i\in I}$ with $\biguplus_{i\in I}S_i\in\mathcal S$, then we even got equality in $(1)$.
So, we immediately obtain that if $\mu$ is ($\sigma$-)additive, then $\left.\operatorname{Var}_\mu^{\mathcal S}\right|_{\mathcal S_{\text{loc}}}$ is ($\sigma$-)additive. And clearly, if $\mathcal S$ is $\cap$-stable, then $\mathcal S_{\text{loc}}=\mathcal S$.
Let $\mathcal A:=\sigma(\mathcal S)$ denote the $\sigma$-algebra generated by $\mathcal S$. Suppose $\mu$ has an extension to a $\sigma$-additive $\nu:\mathcal A\to E$. What is the relation between $\operatorname{Var}_\mu^{\mathcal S}$ and $\operatorname{Var}_\nu^{\mathcal A}$?
My hope is that $\operatorname{Var}_\mu^{\mathcal S}$ and $\operatorname{Var}_\nu^{\mathcal A}$ at least coincide on $\mathcal S$.
If necessary, you can assume that $\mathcal S$ is a ring or even an algebra on $\Omega$.
I've got a pretty hard time to figure out under which conditions on the objects we are able to prove the existence of a unique extension $\nu$ as described above.
I've consulted Theorem 7.4 of Vector Integration and Stochastic Integration in Banach Spaces by Dinculeanu and Theorem 5.2 of Vector Measures by Diestel and Uhl.
My problem with Dinculeanu is his notation for the total variation of a measure, since that notation doesn't make clear from which system ($\mathcal S$ or $\mathcal A$) the constituents of the partitions in the supremum are taken.
So, I'm interested in a further reference for the extension of $\mu$. The case I'm most interested in is $\mathcal S$ being a ring, $E$ being complete and $\mu$ being $\sigma$-additive (on $\mathcal S$) with $\operatorname{Var}_\mu^{\mathcal S}(\Omega)<\infty$.
