Dambis-Dubins-Schwarz proof of the Novikov condition

Write $M_t=B_{[M{]}_t}$ by the Dambis-Dubins-Schwarz theorem and stop the exponential Brownian martingale when $B_s-s$ first reaches $b<0$. The stopped exponential has mean one. Its boundary contribution before $[M{]}_T$ is at mostwhich vanishes as $b\to -\infty$. The remaining term converges monotonically to $\mathcal{E}(M{)}_t$, proving that its expectation is one.