Suppose $\pi$ is a prior distribution on $\Theta$ such that the Bayes risk of the Bayes rule equals $\sup_{\theta\in \Theta}R(\delta_\pi,\theta)$, where $R(\delta,\theta)$ is the risk function associated to the decision problem. Prove that $\delta_\pi$ is least favourable.
Okay this means that $\sup_{\theta\in \Theta}R(\delta_\pi,\theta)=\int_\Theta R(\delta_\pi,\theta)\pi(\theta)\,d\theta\le\int_\Theta R(\delta',\theta)\pi(\theta)\,d\theta$ for any $\delta'$, and we need to show that $\int_\Theta R(\delta_\pi,\theta)\pi(\theta)\,d\theta\ge\int_\Theta R(\delta_\lambda,\theta)\lambda(\theta)\,d\theta$ for any prior distribution $\lambda$.