Martingale is preserved under enlargement to a filtration with usual conditions

Question

Let $\left\{ M_{t},\mathscr{H}_{t}:0\le t<\infty\right\}$ be a continuous martingale on $\left(\Omega,\mathscr{F},P\right)$. If we define $\mathscr{\widetilde{H}}_{t}={\displaystyle \bigcap_{s>t}\sigma\left(\mathscr{H}_{s}\cup\mathscr{N}\right)}$, where $\mathscr{N}$ is the collection of $P$-negligible events in $\mathscr{F}$, then the filtration $\left\{ \mathscr{\widetilde{H}}_{t}\right\}$ satisfies the usual conditions. Do we have $\left\{ M_{t},\mathscr{\widetilde{H}}_{t}:0\le t<\infty\right\}$ is a martingale? What if $M$ is a continuous local martingale?

Answer 1

I found this result. For readers, please refer to C. G. Rogers and David Williams, Diffusions, Markov Processes, and Martingales, Vol I, Page 173, Lemma 67.10.