 Solution

We may take the everywhere-defined representativeIt agrees with the likelihood from part a, hence certainly agrees ${\mu }_0\otimes {\lambda }_k$-almost everywhere. For every $u$, it is a strictly positive density in $y$ and is continuous in $y$. Moreoverso the constant function $h(u)=1$ is an integrable dominator for every probability measure ${\mu }_0$. All four sufficient assumptions from part 1d therefore hold, and the Bayesian inverse problem is well posed in total variation distance.