I have bivariate normal prior distribution over $[x_1 x_2]$:
$$ \begin{bmatrix}x_1 \\ x_2 \end{bmatrix} \sim N\left(\begin{bmatrix} \mu_1 \\ \mu_2 \end{bmatrix}, \Sigma\right),$$ and conditional distribution $y|x_1, x_2$: $$ y | x_1, x_2 \sim N(x_1+x_2,\sigma^2).$$
I want to derive the reverse conditional distribution $x_1, x_2 | y$ as $\sigma \rightarrow 0$, ie when we have deterministic $y=x_1+x_2$. In other words, I want to write the conditional distribution as a function of $\sigma$, and then show that it converges to the standard result at $\sigma=0$. Is it possible to show that this function converges?