Full Reflection at a Measurable Cardinal

Abstract

A stationary subset $S$ of a regular uncountable cardinal $\kappa$ {\it reflects fully} at regular cardinals if for every stationary set $T\subseteq \kappa$ of higher order consisting of regular cardinals there exists an $\alpha \in T$ such that $S\cap \alpha$ is a stationary subset of $\alpha$. {\it Full Reflection} states that every stationary set reflects fully at regular cardinals. We will prove that under a slightly weaker assumption than $\kappa$ having Mitchell order ${\kappa }^{++}$ it is consistent that Full Reflection holds at every $\lambda \le \kappa$ and $\kappa$ is measurable.