A well-known result of Solovay states that $ZFC$ + "the continuum is real-valued measurable" is equiconsistent with $ZFC$ + "there is a measurable cardinal", over $ZFC$. That the continuum is mentioned here is no coincidence. A theorem of Ulam back in 1930 gave evidence that the continuum is some sort of a dividing line: any real-valued measurable cardinal above the continuum must be measurable. On the other hand, cardinals at or below the continuum cannot be measurable (or even strongly inaccessible), and in general are a lot more tame. 


A natural question to follow-up would be, where can the least real-valued measurable cardinal lie, using the continuum as a benchmark? In particular, I would like to ask if each of the following is known to be consistent with $ZFC$ in conjunction with any large cardinal axiom:


1. the least real-valued measurable cardinal $< \mathfrak{c}$.
2. the least real-valued measurable cardinal $= \mathfrak{c}$.
3. the least real-valued measurable cardinal $> \mathfrak{c}$.


Solovay's result means that either 1. or 2. is consistent, but I cannot find any hint as to which of the two holds in his model.



Consistency of the size of the least real-valued measurable cardinal, vis-a-vis the continuum
All of them are equiconsistent with the existence of a measurable cardinal. I find that a nice reference for this stuff, in addition to Solovay's article, is Jech's *Set Theory: Third Millennium Edition*. 


I will discuss 3 first. Any measurable cardinal is real-valued measurable (it just happens not to use any of the values between 0 and 1). Solovay proved that if $\kappa$ is real-valued measurable, $\mu$ is a normal measure on it with $I$ the ideal of null sets, then in $L[I]$ the cardinal $\kappa$ is measurable. Section 3.6 of his paper has a reference to a paper by Kunen proving that the generalized continuum hypothesis holds in $L[I]$, so $\mathfrak{c} = \aleph\_1$ is not real-valued measurable in $L[I]$ (nor is any smaller cardinal).


Case 2 is what occurs if we do Solovay's construction of adding $\kappa$ random reals, where $\kappa$ is a measurable cardinal (which becomes an atomlessly real-valued measurable in the forcing extension).


Case 1 happens if we take a $\lambda > \kappa$ where $\lambda$ has uncountable cofinality (this is automatic if $\kappa = \lambda$ itself because it is regular, but the extra assumption is needed here), and force $\lambda$ random reals. In the forcing extension, $\mathfrak{c} = \lambda$ and $\kappa$ is atomlessly real-valued measurable, and $\kappa < \lambda$ is unchanged. This, and case 2, are explained in section 4 of Solovay's paper, and also in Chapter 2 of Jech's book, around page 415. 


