Integrable-supremum martingale criterion
= Integrable-supremum martingale criterion
{title2=$\mathbb E\sup_{s\leq T}|M_s|<\infty$}
If a <local martingale> has an integrable running absolute supremum on each fixed finite horizon, localization and conditional dominated convergence make it a true <martingale>. This upgrades polynomial Itô local martingales when maximal moments and bracket moments control all terms.