I am working on a bayesian framework where I place a Gaussian Process on my function $f\sim GP$ and have data $D^n=\{(X_i,Z_i,W_i)\}^n$.
I then have the posterior measure $\mu(f|D^n)$. The posterior mean estimator is given by $\hat{f}=\int fd\mu(f|D^n)$.
I now am interested in finding the error of another estimator which i define as: \begin{align*} \beta(x,z) &=E_W[\hat{f}(x,z,W)]\\ &=\int_\mathcal{W}\hat{f}(x,z,W)dP(W)\\ &=\int_\mathcal{W}\int_\mathcal{F}f(x,z,W)d\mu(f|D^n)dP(W) \end{align*}
I am now wondering if I can invoke Fubini Tonelli to switch order of integration $\int_\mathcal{W}\int_\mathcal{F}f(x,z,W)d\mu(f|D^n)dP(W)=\int_\mathcal{F}\int_\mathcal{W}f(x,z,W)dP(W)d\mu(f|D^n)$.
I guess my doubt comes from the fact that I am not integrating against the entire data distibution $P(X,Z,W)$ and the measure $\mu$ is data dependent.
