Measure changes with extinction

We consider a martingale change of measure Q|ℱt≔ZtP|ℱtQ|ℱt≔ZtP|ℱt and clarify that in general 1/Zt1/Zt is only a supermartingale under QQ. Defining Υ≔inf{t≥0:Zt=0}Υ≔inf{t≥0:Zt=0} as the extinction time, we then give a necessary and sufficient condition under which the identity P(Υ=∞)=P(Z∞>0)=Q[1/Z∞]P(Υ=∞)=P(Z∞>0)=Q[1/Z∞] holds.