Let

- $\Omega$ be a set
- $\mathcal A$ be a $\sigma$-algebra on $\Omega$
- $\mu:\mathcal A\to\mathbb R$ be $\sigma$-additive

If $\mathcal S\subseteq2^\Omega$ with $\emptyset\in\mathcal S$, then $$\operatorname{Var}_\mu^{\mathcal S}(A):=\sup\left\{\sum_{i=1}^n|\mu(S_i)|:n\in\mathbb N\text{ and }S_1,\ldots,S_n\in\mathcal S\text{ are pairwise disjoint with }\biguplus_{i=1}^nS_i\subseteq A\right\}$$ for $A\subseteq\Omega$.

In Corollary 1.10 of the book

Vector Measuresby Diestel and Uhl, the authors seem to use the identity $$\operatorname{Var}_\mu^{\mathcal S}(S)=\operatorname{Var}_\mu^{\mathcal A}(S)\;\;\;\text{for all }S\in\mathcal S\;.\tag1$$ In their scenario, $\mathcal S$ is an algebra on $\Omega$ with $\sigma(\mathcal S)=\mathcal A$. How can we prove $(1)$?

I don't know how we need to argue, but if $(\mu^+,\mu^-)$ denotes the Jordan decomposition of $\mu$, then we clearly have $$\operatorname{Var}_\mu^{\mathcal A}(A)=(\mu^++\mu^-)(A)\;\;\;\text{for all }A\in\mathcal A\tag2\;.$$

In the same way we obtain the following: If $\mathcal S$ is $\cup$-stable, then $$\operatorname{Var}_\mu^{\mathcal S}(S)\le(\mu^++\mu^-)(S)\;\;\;\text{for all }S\in\mathcal S\tag3\;.$$ The problematic thing is the other inequality.

The other inequality in $(2)$ is proved in the following way: Let $(\Omega^+,\Omega^-)$ be a Hahn decomposition of $\Omega$ with respect to $\mu$. Then, $$A\cap\Omega^\pm\in\mathcal A$$ are disjoint with $$A=A\cap\Omega^+\uplus A\cap\Omega^-\tag4$$ and hence \begin{equation}\begin{split}(\mu^++\mu^-)(A)&=\mu(A\cap\Omega^+)-\mu(A\cap\Omega^-)\\&=|\mu(A\cap\Omega^+)|+|\mu(A\cap\Omega^-)|\\&\le\operatorname{Var}_\mu^{\mathcal A}(A)\end{split}\tag5\end{equation} for all $A\in\mathcal A$. The key point is that $(4)$ is a partition of $A$ with elements from $\mathcal A$. Since we don't know whether $A\cap\Omega^\pm\in\mathcal S$, we cannot proceed in the same way to establish $(1)$.