I am not 100% certain this question is appropriate for MO; I may just be missing something obvious. Also, I vaguely recall a similar question being asked here a while ago, but I can't find it; if it turns out this is a duplicate, I'll delete this question. Anyways, apologies in advance if this is too easy or is a duplicate. Note that e.g. Can measures be added by forcing? prevents the obvious nuke from working.
Also, the "descriptive-set-theory" tag is purely a guess on my part, based on the surprising ubiquity of descriptive set theory in similar-sounding questions.
Suppose I have a transitive model $M$ of $ZFC$, and - in $M$ - $U$ is a measure on $\kappa$. Then the transitive collapse of the ultrapower of $M$ along $U$ is an inner model, $N\subset M$.
My question is:
Can we ever have $M$ be a generic extension of $N$ (either by set or class forcing in $M$)?
As mentioned above, I am almost certain the answer is "no", even if $M$ has loads of large cardinals, but I don't see how to prove this.
EDIT: Douglas Ulrich answered the question for set forcing; I've asked the class forcing version as a separate question.
A small observation:
Say (inside a model $W$) a cardinal $\mu$ is
potentially measurable if $\mu$ is measurable in some forcing extension; and
reversibly measurable if $\mu$ is measurable, and $W$ is a forcing extension of the transitive collapse of the ultrapower of $W$ by a measure on $\mu$ (that is, if $\mu$ is as above).
Then suppose we had such an $M, N, U, \kappa$, with $j$ the elementary embedding. Then $N$ satisfies "There is a potential measurable below $j(\kappa)$," so - pulling back along $j$ - $M$ satisfies "There is a potential measurable below $\kappa$." This shows that - in $M$ - the least potentially measurable is strictly less than the least reversibly measurable (otherwise we get a descending chain of measurable cardinals).
Now, it feels plausible to me that there's a clever trick that can be done here to outright build a descending sequence of reversibly measurables from a single reversibly measurable; but I don't see it.