Let $u$ be a measure sequence derived from some embedding $j:V\rightarrow M$. I heard that from iterating $j$ is possible to define a generic filter for some Radin forcing $\mathbb{R}\_w$. Specifically, $\mathbb{R}\_w$ is the Radin forcing defined using the measure sequence $w=j\_{\alpha}(u)$, where $j\_\alpha$ is the $\alpha^{th}$-iterated of $j$.


**My question is the following:** Which are the measures that we use to carry out this process? Are the second coordinates (i.e. $u\_\gamma(1)$) of the corresponding measure sequence $u\_\gamma$? Are the whole measure sequences $u\_\gamma$? In this latter case, how looks like the definition of a ultrapower given by a measure sequence $u\_\gamma$?



Radin generics from iterated ultrapowers
See section 5 (called "Generating generic sequences by iterating j") of Radin's paper [Adding closed cofinal sequences to large cardinals](https://www.sciencedirect.com/science/article/pii/0003484382900237?via%3Dihub).


**More details:** The iteration is the usual one, using $j$ (see my comment below). Now the point is that is $u$ is the measure sequence in $V$, then there is $\sigma: ORD \to V$ so that $\sigma(\alpha)=j\_{0, \alpha}(u)\restriction \beta\_\alpha,$ for some suitable $\beta\_\alpha$ and such that for any $\alpha, \sigma \restriction \alpha+1$ is $\mathbb{R}\_{\sigma(\alpha)}$-generic over $M\_\alpha$.


