Finitely additive measures on $\mathbb Z$ and additive combinatorics

We study (bounded) finitely additive measures on the group of integers $\mathbb Z$, as elements of the Banach algebra $\mathrm{ba}(\mathbb Z)$, viewed as a natural generalization of ultrafilters. The algebraic structure of $\mathrm{ba}(\mathbb Z)$ extends the semigroup structure of the \v{C}ech--Stone compactification, allowing methods from ultrafilter theory to be applied in a broader measure-theoretic setting. We investigate idempotent finitely additive measures and establish additive properties of subsets of $\mathbb Z$ having positive measure. We then proceed to study almost translation-invariant and translation-invariant finitely additive measures, showing that these stronger notions yield correspondingly stronger additive conclusions. In particular, we prove that every subset of $\mathbb Z$ whose measure exceeds a certain explicit threshold necessarily is an $\mathsf{IP}_{n}$-set; with stronger properties and lower thresholds depending on the properties of the relevant measures. Several examples illustrating the sharpness and limitations of the results are also presented, together with a discussion of open problems and directions for future research.