= Solution
{c}
Let $\pi_0$ be the <probability density function> of the unknown and $\rho$ that of the noise, both relative to <Lebesgue measure>. By <independence of random variables>, the <conditional distribution> of the measurement given $u$ has density $\rho(m-Au)$. Thus its <likelihood function> and the joint density are
$$
\pi(m\mid u)=\rho(m-Au),\qquad
\pi(u,m)=\pi_0(u)\rho(m-Au).
$$
The <Bayesian model evidence> is the marginal data density
$$
Z(m)=\int_{\mathbb R^d}\pi_0(v)\rho(m-Av)\,dv.
$$
<Bayes' theorem> gives the <posterior density> solving this <Bayesian inverse problem>:
$$
\boxed{\pi^m(u)=\frac{\pi_0(u)\rho(m-Au)}{Z(m)}.}
$$
This formula applies when $0<Z(m)<\infty$. Since $\int Z(m)\,dm=1$ by <Tonelli theorem>, these conditions hold <almost everywhere> under the marginal data law. Values of a <conditional distribution> at exceptional data are not determined by the joint law. The <posterior mean> or a <maximum a posteriori estimate> can then give a point estimate, while the full <posterior density> describes the remaining uncertainty.
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