 Martingale transfer under a density process

Let $Z$ be a positive uniformly integrable martingale defining $d\tilde{\mathbb{P}}=Z_{\infty }d\mathbb{P}$. If $Y$ is bounded and $ZY$ is a true $\mathbb{P}$-martingale, then $Y$ is a $\tilde{\mathbb{P}}$-martingale. This follows from the conditional-expectation change-of-measure identity.