= Solution
A <Bayesian inverse problem> consists of a <prior distribution> $\mu_0$ on the unknown $u\in X$, a reference measure on the data space $Y$, and a jointly <measurable function>[measurable] likelihood $L(u;y)$ such that $L(u;\mathord\cdot)$ is a <probability density function> for $\mu_0$-almost every $u$. For observed data $y$, <Bayes theorem> defines the <posterior distribution> by
$$
\boxed{
\frac{d\mu^y}{d\mu_0}(u)
=\frac{L(u;y)}{Z(y)},
\qquad
Z(y)=\int_XL(u;y)\,d\mu_0(u)},
$$
provided $0<Z(y)<\infty$.
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