Background: I am reading a proof of Girsanov's theorem, it goes:
"Lemma: The density process $M_t=\exp(\lambda B(t)-\frac{1}{2}\lambda^2 t)$ is a P-martingale. Proof: Applying Itô's formula gives us: \begin{align} \mathrm{d}M_t &= M_t\lambda\mathrm{d}B_t - \frac{1}{2}M_t\lambda^2\mathrm{d}t + \frac{1}{2}M_t\lambda^2\mathrm{d}t\\ &= M_t\mathrm{d}B_t. \end{align} It then follows from the Martingale representation theorem that $M_t$ is a martingale."
Question: I understand that it is a matringale, but I cant see that it follows from MRT. So my question: If we have a stochastic process $$Z_t = Z_0 + \int_0^t \lambda Z_s\mathrm{d}B_s,$$ can we then from the Martingale representation theorem deduce that $Z_t$ is a martingale somehow? What conditions must we impose on $Z$?
I think the author may have misspoke, we do know that if $\xi$ is $B$ integrable, then $\int \xi ~ dB$ is a martingale. For example see here. The martingale representation theorem states the converse of this.