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:

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