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.

Let $Z$ be a positive uniformly integrable density martingale. If $Y$ is bounded and $ZY$ is a local martingale under the original measure, the product is a true martingale: its stopped absolute values are dominated by a fixed multiple of the uniformly integrable stopped density. Bayes then makes $Y$ a martingale under the new measure. Stop a locally bounded continuous $Y$ at absolute-value levels to obtain the local version.