a measurable cardinal  & a real-valued measurable cardinal in the same model? Although I know that "ZFC & there exists a measurable cardinal" and "ZFC & there exists a real-valued measurable cardinal" are equiconsistent with one another, I am not sure whether "ZFC & there exists a measurable cardinal k & there exists a real-valued measurable cardinal b" is equiconsistent with ZFC. (Obviously k is not equal to b.)
I would be grateful for an answer. 
 A: Let me interpret the question as asking for a real-valued measurable cardinal that is not measurable, plus another (two-valued) measurable cardinal, which must be above it. For example, in a more extreme form, your question would ask: can the continuum be real-valued measurable while there is also another measurable cardinal? 
The answer is yes.
Theorem. If there are two measurable cardinals, then there is a forcing extension in which the smaller one becomes the continuum and real-valued measurable, and the larger one remains measurable. 
Proof. If $\kappa\lt\lambda$ are both measurable cardinals in $V$, then the forcing to add $\kappa$ many random reals will make $\kappa$ into the continuum and still real-valued measurable (by a result of Solovay), and the larger measurable cardinal $\lambda$ remains measurable, by the Levy-Solovay theorem, because the forcing was much smaller than $\lambda$. QED
Corollary. The following are equiconsistent.


*

*There are two measurable cardinals. 

*There are two real-valued measurable cardinals. 

*There is a non-measurable real-valued measurable cardinal and a measurable cardinal. 

*The continuum is real-valued measurable and there is another measurable cardinal. 


Proof. Statement 1 implies 4 in a forcing extension, by the argument above. Statement 4 implies statement 3 directly. Statement 3 implies statement 2 directly. Statement 2 implies statement 1 in an inner model. QED
Of course, one can generalize the arguments to handle any number of real-valued measurable cardinals--three instead of two, or any cardinal number---with measurable cardinals above them.
