A paper I read recently seems to make the following statement: if $\gamma_t$ is a progressively measurable process, and that $\exp\left(\int_0^T\gamma_s dW_s\right) \in L^p$ for some $p>1$, then the exponential local martingale $$\exp\left(\int_0^T\gamma_sdW_s - \frac{1}{2}\int_0^T\gamma^2_s ds\right)$$ is a true martingale.
This was not stated explicitly as a result. Instead $\exp\left(\int_0^T\gamma_s dW_s\right)$ is assumed to be in $L^p$ and the author applied the change of measure without further explanation.
Does it actually hold true that $L^p$ integrability of $\exp\left(\int_0^T\gamma_s dW_s\right)$ implies the martingality of $\mathcal{E}\left[\int \gamma_s dW_s\right]$? And if so, can you please give a reference for the result?
Thanks!