 Bounded-increment optional stopping with integrable time

For an integrable supermartingale with the displayed increment bound, $|X_{T\wedge n}|\le |X_0|+KT$. Thus the stopped values are dominated by one integrable random variable, including $X_T$. Apply optional stopping at the bounded times $T\wedge n$ and then the dominated convergence theorem. For a martingale the same proof gives equality.