# real-valued-measurable cardinals and the cardinality of the continuum

• February 6th 2011, 02:19 AM
(1) "Ulam also showed that successor cardinals like $\aleph_1$ cannot be real-valued measurable."
(2) "Solovay …… showed that if $\kappa$= $2^{\aleph_0}$ is real-valued measurable then there is an inner model (namely L[I] where I is the ideal of null sets) wherein $\kappa$ is still real-valued measurable and GCH holds."