This is a follow-up question to [this one](https://mathoverflow.net/questions/359488/finitely-additive-kappa-additive-atomless-measures-in-zfc).


Is there a ZFC example of an atomless measure that is $2^\omega$-additive, meaning, fewer than continuum many null sets have measurable union that is null?



Atomless, c-additive measures in ZFC
The answer is negative. Suppose $(X,\mu)$ is such a measure space. By the argument in the linked question, there is a partition of $X$ into continuum-many pairwise disjoint $\mu$-null sets. This is done by building a binary tree that splits a given node into two nodes of one half the measure. Each branch corresponds to a point in Cantor space. We induce an atomless $2^\omega$-additive measure $\nu$ on Cantor space via this correspondence.


However, it is consistent with ZFC that the continuum is singular. Under this hypothesis, any $2^\omega$-additive ideal is $(2^\omega)^+$-additive. But each singleton is $\nu$-null, and so the Cantor space is the union of $2^\omega$-many $\nu$-null sets, meaning the whole space has measure 0. But this means that $\mu(X)=0$.


