I was reading a chapter in Dellacherie and Meyer. Suppose we have right continuous adapted processes $A$, $A'$ of finite variation. Both are null at zero and the difference is a local martingale.
I know the following lemma:
A continuous local martingale of finite variation is constant.
In Dellacherie and Meyer they conclude that in the above situation $A=A'$. So can continuity in the above lemma be relaxed to right continuity?
Thanks in advance.