I encountered the following in a proof when studying Bayes.
Let $\mathbb{Q}_0(dy)$ denote the $\mathbb R^m$-valued Gaussian measure $N(0,\Sigma)$ and $\mathbb{Q}(dy\mid x)$ the $\mathbb R^m$-valued Gaussian measure $N(\mathcal{G}(x), \Sigma)$. By construction $$\frac{d\mathbb{Q}}{d\mathbb{Q}_0}(y\mid x)=\exp\left(-\frac{1}{2}|y-\mathcal{G}(x)|_{\Sigma}^2+\frac{1}{2}|y|_\Sigma^2\right) $$
I tried to understand the notations. But I could not get the $\mathbb{Q}_0(dy)$ and $\mathbb{Q}(dy\mid x)$. What does $dy$, $dy\mid x$ mean inside the $\mathbb{Q}$ and how does it contribute to the construction.
What if I have $\mathbb{Q}_0(dx)$, do I get $\frac{d\mathbb{Q}}{d\mathbb{Q}_0}(y\mid x)=\exp(-\frac{1}{2}|y-\mathcal{G}(x)|_{\Sigma}^2+\frac{1}{2}|x|_\Sigma^2) $. Is it possible to derive the construction and deal with the derivatives step by step like in calculus?
