Solution

The following four assumptions are sufficient, with statements understood for

μ_{0}

-almost every

u

and every

y

$L (u; \cdot)$ is a strictly positive probability density function on the data space.
$L (\cdot; y) \in L^{1} (X, μ_{0})$ .
There is one $h \in L^{1} (X, μ_{0})$ such that $L (u; y) \leq h (u)$ for all $y$ .
For each relevant $u$ , the map $y \mapsto L (u; y)$ is continuous.

The first two assumptions give

0 < Z (y) < \infty

. The last two allow the dominated convergence theorem to prove

L^{1} (μ_{0})

continuity of the normalized posterior density, which is equivalent to continuity in total variation distance.

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