For $\kappa$ a measurable, and $U$ is (normal) measure, we know that $L[U]\models\mathsf{GCH}$. That $\mathsf{GCH}$ holds above $\kappa$ is easy: what I'm interested in is $\mathsf{GCH}$ below $\kappa$. Most of the regular proofs of this fact resort properties of Rowbottom cardinals. I've been told it can be done with the comparison method/process, but I'm not sure how this argument would go.
My intuition would be to proceed as in $L$: for $x\subseteq\lambda<\kappa$, take a skolem hull and get some $L_\alpha[W]$, and then iterate until things match up, and then argue somehow about the relation between $U$ and the resulting embedded ultrafilter $W_\gamma$. The end goal would be that we get $x$ in some $L_\beta[U]$ for $\beta<\lambda^+$, but it's not obvious to me how to show this if this is even the right route to take.
Any reference or advice would be appreciated!