A cardinal $\kappa$ is *real-valued measurable* if there is a $\kappa$-additive probability measure on $2^\kappa$ which vanishes on singletons. The existence of measurable $\kappa$ is independent of ZFC.


Question: if $\kappa$ is assumed to be real-valued measurable, does it necessarily follow that $2^\kappa$ is real-valued measurable?



Does measurability of cardinal $\kappa$ imply measurability of $2^\kappa$?
No, this is not necessarily true. In fact, this is almost never true - it can hold only if $2^\kappa$ is equal to the continuum.


Indeed, assume $2^\kappa$ is strictly greater than the continuum. By standard results (see e.g. [Wikipedia](https://en.wikipedia.org/wiki/Measurable_cardinal#Real-valued_measurable)), a cardinal greater than continuum is real-valued measurable iff it's measurable. But measurable cardinals are necessarily strong limits, and $2^\kappa$ is, quite blatantly, not one.


