= Solution
The following four assumptions are sufficient, with statements understood for $\mu_0$-almost every $u$ and every $y$:
* $L(u;\mathord\cdot)$ is a strictly positive <probability density function> on the data space.
* $L(\mathord\cdot;y)\in L^1(X,\mu_0)$.
* There is one $h\in L^1(X,\mu_0)$ such that $L(u;y)\leq h(u)$ for all $y$.
* For each relevant $u$, the map $y\mapsto L(u;y)$ is <continuous function>[continuous].
The first two assumptions give $0<Z(y)<\infty$. The last two allow the <dominated convergence theorem> to prove $L^1(\mu_0)$ continuity of the normalized posterior density, which is equivalent to continuity in <total variation distance>.
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