proposition. A permanent barrier requires transition closure [ftip-00MR]

Let \(\mathcal R\) be a measurable set of complete states. Suppose the initial state lies in \(\mathcal R\) almost surely, and every admitted transition from a permitted history ending in \(\mathcal R\) remains in \(\mathcal R\) with probability one. Suppose no successful terminal state lies in \(\mathcal R\). Then for every admitted campaign,

\[\Pr (T_{\mathrm {success}}<\infty )=0.\]