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  • $\begingroup$ Regarding 1: If by "real-valued measurable" we mean atomlessly measurable (rather than measurable in the usual sense of large cardinals), then any real-valued measurable cardinal is of size at most the continuum. $\endgroup$
    – Andrés E. Caicedo
    Commented May 2, 2018 at 23:31
  • $\begingroup$ Can you briefly illustrate how adding $\kappa$ random reals in Case 2 would not make any cardinal below $\kappa$ real-valued measurable? $\endgroup$
    – Zoorado
    Commented May 3, 2018 at 2:42
  • 5
    $\begingroup$ @Zoorado If you start with $L[\mu]$, the smallest model with a measurable cardinal $\kappa$ (with $\kappa$ as small as possible) and add $\kappa$ random reals, Solovay's argument shows that in the extension $\kappa$ is real-valued measurable and the continuum. If $\lambda<\kappa$ is also real-valued measurable, Solovay's argument using the null ideal for a witnessing measure on $\lambda$ would give you an inner model with $\lambda$ measurable. This contradicts the minimality of $\kappa$. (The one technical further detail is that minimality of $L[\mu]$ is absolute under forcing extensions.) $\endgroup$
    – Andrés E. Caicedo
    Commented May 3, 2018 at 3:24