This is possibly a reference request. Let $G$ : $\mathbb{R}^p \to \mathbb{R}^q$ be a continuous injective/bijective function. Let $\mu$(we may also assume this to be a non degenerate Gaussian) be probability measure(prior) on $\mathbb{R}^p$ which is absolutely continuous wrt to Lebesgue measure. Let the observation model be $$y = G(u) + \frac{\eta}{\sqrt{n}}$$ Where $\eta$ is the white noise on $\mathbb{R}^q$. Does the posterior measure $\mu_n^y$(which is the ditribution of $u|y)$ converge to the given true solution $u_0$ for almost all $u_0$ wrt the measure $\mu$?
The convergence condition for a true solution $u_0$ is defined as follows. Let $\mathbb{E}_{u_0,n}$ denote expectation with respect to the distribution of random variable $G(u_0) + \frac{\eta}{\sqrt{n}}$. $\mu_n^y$ is said to converge to $u_0$ if $$\mathbb{E}_{u_0,n}\left(\mu_n^y\{u:\|u -u_0\| > \epsilon\}\right) \to 0$$ as $n \to \infty$
