**Edit:** I simplified the example to a canonical case for clarity.

Given an integral $\int_{\Omega}{g(\mathbf{x})}$ with a well-posed integrand $g(\mathbf{x})$ defined on some multidimensional space $\Omega$, one can integrate it successfully with a Markov Chain Monte Carlo method, in particular using Metropolis-Hastings method. Detailed balance and ergodicity are achieved with any well-posed (samplable) integrand. Transition kernel $K(\dot{},\dot{})$ is Harris recurrent and its transition probability is known for every pair $(X,Y)$. The value of the desired integral is known to always exist and be finite, even in presence of delta distributions in the integrand. This is given.

In the case of interest, the integrand is given as $f(\mathbf{x})=\delta_{\mathbf{x}_0}(\mathbf{x})\dot{} g(\mathbf{x}) + h(\mathbf{x})$ and consists of a delta distribution at some unknown location $\mathbf{x}_0\in \Omega$ and some regular (well-posed) non-zero functions $g(\mathbf{x})$ and $h(\mathbf{x})$. This delta distribution cannot be sampled explicitly or using numerical optimization (given). This makes such an integrand unsamplable with random walk or probabilistic sampling. 

I mollify (approximate to the identity) this delta distribution using some mollifier (normalized smooth function $\phi_\epsilon(\mathbf{x})=\epsilon^{-1}\phi\left(\frac{\mathbf{x}}{\epsilon}\right)$ with some bandwidth $\epsilon$). This leads to a tempered integrand $f_\epsilon(\mathbf{x})=\phi_\epsilon(\mathbf{x}-\mathbf{x}_0)\dot{} g(\mathbf{x})+h(\mathbf{x})$. During the integration, at every step $n$, I gradually shrink the parameter $\epsilon_n$ to zero in order to achieve $f_{\epsilon_n} \to f$ as $n \to \infty$ in spirit of serial tempering and simulated annealing. Thus I expect the integration method to be consistent, i.e. to give the proper answer in the limit.

Unfortunately one cannot use the usual parallel or serial tempering here, as the probability of the proposal to descend from a tempered mixture to the original mixture $f$ at the exact location $\mathbf{x}_0$ of the delta distribution is zero.

Thus I have two rather similar questions: 

 1. Would the integral converge to the proper value $g(\mathbf{x_0})+\int_{\Omega}{h(\mathbf{x})}$? 
 2. What are the conditions for the asymptotic decrease rate of the sequence $\{ \epsilon_n \}$, in order to guarantee the consistent convergence of the MCMC estimate? In other words, that allows the integral to converge before parts of the integrand $f_{\epsilon_n}$ become unsamplable?