Consider a Bayesian decision problem in which $F$ is the prior distribution over some parameter space $T$. If $- \psi (d,t)$ is the product of the loss function and the likelihood function, then the Bayesian solution, $d_{F}$, maximizes $E_{F} (d) = \int_{T} \psi (d, t) dF (t)$. Suppose $\{F^{n} \}$ is a sequence of distribution functions that approach $F^{0}$ in the sup-metric topology. Our main theorem gives conditions under which $d_{F^{n}} \rightarrow d_{F^{0}}$ and E_{F^{0}}(d_{F{^n}}) \rightarrow E_{F^{0}} (d_{F^{0}})$.