Solovay proved that from ZFC + existence of an inaccessible cardinal there is a model of ZF + DC + LM, where LM is "every set of reals is Lebesgue measurable."
If we instead start from ZFC + existence of $n$ inaccessible cardinals, does a construction like Solovay's give a model of ZF + DC + LM + existence of at least $n-1$ inaccessible cardinals?
I believe the answer is yes, but I am a complete novice in this area, so I am unsure. The sketch of an argument I have in mind is the following:
Solovay starts by taking the Levy collapse $V[G]$ for a generic $G$ that collapses all cardinals less than the first inaccessible $\kappa$ to $\omega$. I believe this preserves the other inaccessible cardinals. (Actually, I guess it even preserves $\kappa$?)
One can then pass to the inner model $L(R)$, where the desired result holds. (Kanamori points out that this is different than Solovay's original argument, but that it also works). But inaccessible cardinals are downward absolute so we still have at least $(n-1)$ inaccessible cardinals in $L(R)$. (Right?)
I understand that there are different notions of inaccessibility in a setting without choice, so I guess I should clarify that what I'm really after is something that would give me (several) Grothendieck universe(s). The reason is that it seems that a significant amount of measure theory and analysis can be done with ZF + DC (e.g. as in Fremlin's book), but in some other fields it is a technical convenience to be able to have Grothendieck universes.
Sorry if the question is rather naive/confused. I mainly want to use this result as a black box given the considerable prerequisites (which are admittedly rather fascinating!).
This question seems to be partially addressed by Asaf Karagila's answer to Implications of existence of two inaccessible cardinals? but I'm not completely certain.