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](https://arxiv.org/abs/1607.08505). 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.)



Real-valued measurable cardinals
Suppose $m:\mathcal{P}(\kappa) \to [0. 1]$ is a finitely additive measure whose null ideal $I$ is $\kappa$-additive. Then $I$ is an $\omega\_1$-saturated $\kappa$-additive ideal over $\kappa$. Solovay showed this implies that $\kappa$ is measurable in $L[I]$.


If we start with a measurable cardinal $\kappa$, and force with a finite support iteration of random forcing of length $\kappa$, then in the resulting model there is a finitely additive measure on $\mathcal{P}(\kappa)$ whose null ideal is $\kappa$-additive and there is no real valued measurable cardinal below the continuum. You can find a proof of this fact in A. Kumar, K. Kunen, Induced ideal in Cohen and random extensions, Topology and its Applications, Vol. 174, Sept. 2014, 81-87.


