# Stationary vs measurable limits for large cardinals

This is a follow-up to a question I asked earlier over here. Why are stationary limits so ubiquitous when studying large cardinals?

I have noticed that there appears to be a stronger limit notion that appears sometimes - a "measurable limit". From what I understand, a cardinal $$\kappa$$ is a measurable limit of cardinals of type X if there is a normal measure on $$\kappa$$ containing all the cardinals of type X $$< \kappa$$

From this definition, it is unclear to me why a measurable limit is stronger than a stationary limit. For instance, is a measurable limit something like a stationary limit of stationary limits ? Or something much stronger ?

What would be examples of theorems illustrating why a measurable limit is stronger than a stationary limit ?

If there is a normal measure $$U$$ on $$\kappa$$ such that $$A_X\in U$$, where $$A_X$$ is the set of $$X$$-cardinals $$\alpha<\kappa$$, then not only is $$\kappa$$ a limit of inaccessiles $$\kappa'$$ such that $$\kappa'$$ is a stationary limit of $$X$$-cardinals, but in fact the set $$A_{X'}$$ of inaccessible stationary limits of $$X$$-cardinals $${<\kappa}$$ is also in $$U$$.

For let $$j:V\to M$$ with $$\mathrm{crit}(j)=\kappa$$ and $$\kappa\in j(A_X)$$. Then $$M\models$$ "$$\kappa$$ is a stationary limit of $$j(X)$$-cardinals", because $$A_X$$ is stationary in $$V$$ and hence in $$M$$, and in $$M$$, $$A_X$$ is the set of $$j(X)$$-cardinals $$\alpha<\kappa$$. It follows that $$A_{X'}\in U$$.