Two results from the following web-site: Existence of probability measure defined on all subsets - MathOverflow which I am presuming quote the results correctly.

(1) "Ulam also showed that successor cardinals like $\displaystyle \aleph_1$ cannot be real-valued measurable."

(2) "Solovay …… showed that if $\displaystyle \kappa$= $\displaystyle 2^{\aleph_0}$ is real-valued measurable then there is an inner model (namely L[I] where I is the ideal of null sets) wherein $\displaystyle \kappa$ is still real-valued measurable and GCH holds."

But if c is real-valued measurable, that would mean, by (1), that the CH (and hence the GCH) would not hold. So how can you have such a model?