Consider a continuous local martingale $M$ and the stochastic exponential $Z(t)= \exp\{M(t) - 0.5 \langle M \rangle(t)\}$, where $\langle M \rangle(t)$ is the quadratic variation of $M$. Using Ito's formula one can show:
$$dZ(t)=Z(t)dM(t)$$ meaning: $$Z(t)= Z(0)+ \int_0^t Z(s) dM(s)$$
How can one verify that $Z_t$ is local martingale? In general integrals w.r.t. local martingales are local martingales if $Z_s$ is predictable. I cannot see why this is the case here?