Let M be a martingale with respect to the right continuous and complete filtration Then there exists a unique (up to indistinguishability) modification M∗ of M which is cadlag and adapted to {Ft}...

Let M be a martingale with respect to the right continuous and complete filtration

Then there exists a unique (up to indistinguishability) modification M∗ of M which is cadlag and adapted to

{Ft} (hence is an {Ft} martingale).

Suppose M is a martingale bounded in Lp for some p > 1. Then M is uniformly integrable.