With a system in state $|\psi\rangle$ with stabilizer $g_1,...,g_n$, if a measurement $g$ that is commute with all $g_i$ is applied to the state, then either $g$ or $-g$ is a stabilizer of the system and the measurement doesn't disturb the state.
However, it seems to me that there is a simple counterexample:
Let $g$ be the $Z_1$ operator in the system, thus it commutes with all stabilizers as $g|\psi\rangle$ is still a valid state in the system. However neither $g$ nor $-g$ is a stabilizer of the code.
Am I missing anything or the quoted claim is indeed false?