= Solution
For a prior $\Pi$ and loss $L$, the <Bayes risk> of a rule $d$ is $r(\Pi,d)=\int\mathbb E_\theta L(\theta,d(X))\,\Pi(d\theta)$. A <Bayes decision rule> minimizes this integrated risk. An <Extended Bayes rule> is a rule whose excess over the optimal <Bayes risk> can be made arbitrarily small by a suitable proper prior: for every $\varepsilon>0$ some $\Pi$ has $r(\Pi,d)\leq\inf_{d'}r(\Pi,d')+\varepsilon$, with finite risks.
Condition on the observation and use iterated expectation. A rule is Bayes precisely when it chooses a minimizer of the posterior expected loss almost surely, up to the usual measurability requirement. Under squared Euclidean loss, with a finite posterior <second moment>, write $m=\mathbb E[\theta\mid X]$. The identity
$$
\mathbb E[\|\theta-a\|^2\mid X]
=\mathbb E[\|\theta-m\|^2\mid X]+\|m-a\|^2
$$
shows that \b[the Bayes rule is the <posterior mean>].
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