Let us consider that I have two random variables. A discrete one, called $\alpha:\Omega\to\{1,2,3,\ldots,n\}$ and another one called $X:\Omega\to\mathbb{R}$, which is a continuous random variable. I want to compute $\mathbb{E}_{\mathbb{P}}\left[\alpha|X=x\right]$. I know the probability density function of $X$, which I denote by $f_X$. I also know the conditional probability density functions $f_{X|\alpha=i}, (\forall)i\in\{1,2,3,\ldots,n\}$, as well as $\mathbb{P}(\alpha=i), (\forall) i\in\{1.2.3.\ldots,n\}$. With these and availing myself of the Bayes formula, I can compute the desired conditional expectation.
However, let us say that I have a way to generate realizations of X under different measure $\mathbb{Q}$ and I also know $\frac{\mathrm{d}\mathbb{Q}}{\mathrm{d}\mathbb{P}}$. Given a realization $x$ of $X$ drawn from the algorithm that simulated $X$ using measure $\mathbb{Q}$, is it correct to compute the above expectation by simply making the adjustment $x\to x\cdot\left(\frac{\mathrm{d}\mathbb{P}}{\mathrm{d}\mathbb{Q}}\right)$ ? If so, what is the theoretical justification?