I know that for two measures defined on the Borel set of real line $\mu_1$ and $\mu_2$, if $\mu_1((a,b))=\mu_2((a,b))$ for all real numbers a and b, then $\mu_1=\mu_2$.
Curious on whether similar conditions hold for finite-additive, but not countably additive measures. If $v_1(A)$, $v_2(A)$ are two finitely-additive measures and $v_1(A)$ and $v_2(A)$ are equal for all $A \in \beta$, in which $\beta$ is a collection of Borel sets. What is the 'smallest' $\beta$ to ensure that these two measures are equal? You can assume $\mu(\Omega)=1$ as non-standard probability theory if that's necessary.
If it takes a long story to answer, a book/paper name and author is also welcomed. Many thanks for that.
Finitely additive measures are in isometric correspondence with positive functionals over $L^\infty(\Omega)$. Since $L^\infty(\Omega) = C(X)$, where $X$ is the compact Hausdorff topological space given by the Gel'fand spectrum of $L^\infty(\Omega)$, we have that the finitely additive measures are the positive cone of $C(X)^* = M(X)$, where $M(X)$ are the Borel measures over $X$. The set $\beta$ must generate the whole $\sigma$-algebra of $X$, for example this happens when $\beta$ is a base for the topology of $X$.
Example: When $\Omega = \mathbb{N}$, you have that $X = \beta \mathbb{N}$, the Stone-Cech compactification of $\mathbb{N}$ and you can take $\beta$ (confusing notation) to be $\{ \mathbb{N} \cap V : V \in \mathcal{V} \subset \mathcal{P}(\beta \mathbb N) \}$, where $\mathcal V$ is a base for the topology of $\beta \mathbb N$.