Real-valued measurable cardinals and strong ideals

Question

Is it consistent to have (at the same time) a real-valued measurable cardinal and precipitous ideals on small cardinals such as $\omega_1$? What about saturated ideals on small cardinals?

Answer 1

Suppose that there are two measurable cardinals $\kappa\lt\delta$ in $V$. Perform forcing in two steps: first, we form the forcing extension $V[G]$ by the Levy collapse making $\kappa$ become $\omega_1$. The standard arguments (I think due to Prikry) show that $\kappa=\omega_1^{V[G]}$ now carries a precipitous ideal in $V[G]$. Next, we form the extension $V[G][H]$ by pumping up $2^\omega=\delta$ by random real forcing in such a way that $\delta$ is now real-valued measurable, a result due to Solovay. (Both of these arguments are in Kanamori's text.)

Since $V[G][H]$ is a ccc extension of $V[G]$, it follows by a theorem of Kakuda that the precipitous ideal of $V[G]$ generates a precipitous ideal on $\omega_1$ in $V[G][H]$. Thus, in $V[G][H]$ we have both a precipitous ideal on $\omega_1$ and the continuum is a real-valued measurable cardinal.

More generally, if you don't insist that the real-valued measurable cardinal is actually the continuum or less, then all that you need is just a measurable above, and so the forcing $H$ is not required. That is, $V[G]$ already has a precipitous ideal and a measurable cardinal $\delta$, which is also real-valued measurable.

I find it likely, although it would take an inner-model-theory expert to confirm, that the situation you request requires at least two measurable cardinals, because if you have a precipitous ideal on $\omega_1$ and a real-valued measurable cardinal above this, then I expect that there is a fine-structural inner model of ZFC in which both of these cardinals are fully measurable. Perhaps this can be confirmed in another answer by the inner-model-theory experts.