= Solution
We may take the everywhere-defined representative
$$
\boxed{L_0(u;y)=\exp[-2\|y-A(u)\|_1]}.
$$
It 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 $y$. Moreover
$$
0<L_0(u;y)\leq1,
$$
so 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>.
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