To quote from my book, [The Bayesian Choice][1] (2007, Section 2.3, pp. 63-64)

> The Bayesian approach to decision theory integrates on the parameter
> space $\Theta$, since $\theta$ is unknown, instead of integrating on
> the sampling space ${\cal X}$, as $x$ is observed. It relies on the
> ***posterior expected loss*** \begin{eqnarray*} \rho(\pi,d|x) & = &
 \mathbb{E}^\pi[L(\theta,d)|x] \\ \tag{1}
       & = & \int_{\Theta} \mathrm{L}(\theta,d) \pi(\theta|x)\, \text{d}\theta, \end{eqnarray*} which averages the error (i.e., the
> loss) according to the posterior distribution of the parameter
> $\theta$, conditional on the observed value $x$. Given $x$, the
> average error resulting from decision $d$ is actually $\rho(\pi,d|x)$.
> The posterior expected loss is thus a function of $x$ but this
> dependence is not an issue, as opposed to the frequentist dependence
> of the risk on the parameter because $x$, contrary to $\theta$, is known.
> 
> Given a prior distribution $\pi$, it is also possible to define the
> ***integrated risk***, which is the frequentist risk averaged over the
> values of $\theta$ according to their prior distribution
> \begin{eqnarray*} r(\pi,\delta) & = & \mathbb{E}^\pi[R(\theta,\delta)]
  \\ \tag{2}
                & = & \int_{\Theta} \int_{\cal X} \mathrm{L}(\theta,\delta(x))\,
                     f(x|\theta) \,\text{d}x\ \pi(\theta)\, \text{d}\theta. \end{eqnarray*} One particular appeal of this second
> concept is that it associates a real number with every estimator, not
> a function of $\theta$. It therefore induces a total ordering on the
> set of estimator s, i.e., allows for the direct comparison of
> estimators. This implies that, while taking into account the prior
> information through the prior distribution, the Bayesian approach is
> sufficiently reductive (in a positive sense) to reach an
> effective decision. Moreover, the above two notions are equivalent in
> that they lead to the same decision.

A bit further, I use the following definition for the Bayes risk:

> A ***Bayes estimator***  associated with  a prior distribution $\pi$ and a loss
> function $\mathrm{L}$ is any estimator $\delta^\pi$ which minimizes
> $r(\pi,\delta)$. For every $x\in\cal{X}$, it is given by $\delta^\pi(x)$,
> argument of $$\min_d \rho(\pi,d|x)$$ The value $$r(\pi) =
 r(\pi,\delta^\pi)\tag{3}$$ is then called the ***Bayes risk***.

  [1]: http://amzn.to/2kxykkw