  Based on the work of Shelah, Kellner, and T\u{a}nasie (Fund. Math.,
166(1-2):109-136, 2000 and Comment. Math. Univ. Carolin., 60(1):61-95, 2019),
and the recent developments in the third author's master's thesis, we develop a
general theory of iterated forcing using finitely additive measures. For this
purpose, we introduce two new notions: on the one hand, we define a new
linkedness property, called $\mu$-$\mathrm{FAM}$-linked and, on the other hand,
we generalize the notion of intersection number to forcing notions, which
justifies the limit steps of our iteration theory. Our theory also generalizes
iterations with ultrafilters, which have played an important role in the proof
of the consistency of Cicho\'n's maximum.
  We further show that any iteration constructed with our general theory
preserves strong unbounded families and what we call \emph{anti-Bendixson
families}, which play a central role in preserving witnesses of
$\mathrm{cov}(\mathcal{N})$ of singular size (even of countable cofinality).
Finally, we apply our theory to prove the consistency of a new separation of
the left-hand side of Cicho\'n's diagram where $\mathrm{cov}(\mathcal{N})$ is
possibly singular, even with countable cofinality.
