A cardinal $\kappa$ is real-valued measurable if there is a probability measure on the $\sigma$-algebra of all subsets of $\kappa$ which is zero on singletons and additive on disjoint families of fewer than $\kappa$ subsets.
What if I weaken this to: $\kappa$ is uncountable and there is a finitely additive probability measure which is zero on singletons and such that the union of any family of fewer than $\kappa$ null sets is null. Has this notion been studied? What is the consistency strength of the existence of a cardinal with this property?
Edit: A paper has been written on the project which provoked this question. It has been posted on the arXiv. Ashutosh's nice result is discussed in Section 7. (I am adding an "operator algebras" tag because the paper is about large cardinal aspects of von Neumann algebras.)