Saharon Shelah in 2000 introduced a finite-support iteration using finitely additive measures to prove that, consistently, the covering of the null ideal may have countable cofinality. In 2019, Jakob Kellner, Saharon Shelah, and Anda R. Tănasie achieved some new results and applications using such iterations.
In this talk, based on the works mentioned above, we present a general theory of iterated forcing using finitely additive measures, which was developed in the speaker's master's thesis. For this purpose, we introduce two new notions: on the one hand, we define a new linkedness property, which we call "FAM-linked'' and, on the other hand, we generalize the idea of intersection number to forcing notions, which justifies the limit steps of our iteration theory. Finally, we show a new separation of the left-side of Cichoń’s diagram allowing a singular value.