Randomization does not improve convex action loss

Assume the Axiom of Choice and Countable Choice. Let $d\ge 1$ and let the statistical experiment be $\mathsf{E}=((\mathcal{X},\mathcal{A}),\mathrm{\Theta },(P_{\theta }{)}_{\theta \in \mathrm{\Theta }})$, (Statistical experiments), with a nonempty closed convex action set $\mathsf{D}\subseteq {\mathbb{R}}^d$, with its Borel sigma-algebra (Measurable spaces and measurable sets). Closedness is in the Euclidean topology (Interior, closure, boundary, limit point, isolated point and dense subset of a metric space) and convexity uses A convex subset of ${\mathbb{R}}^m$ contains every line segment between two of its points. Let $\delta (x,da)$ be a randomized decision rule, and let $L(\theta ,\cdot ):\mathsf{D}\to [0,+\mathrm{\infty }]$ be a nonnegative Borel loss convex in the extended-functional sense of Convex and strictly convex functionals on a convex subset of a real vector space. On its convex finite-valued domain this inequality agrees with the real-valued Euclidean convention of Convex and strictly convex functions on Euclidean convex sets, used in the Jensen proof's subgradient step. Define $m_{\delta }(x)={\int }_{\mathsf{D}}\mathrm{\parallel }a{\mathrm{\parallel }}_2\delta (x,da),E_0=\{x:m_{\delta }(x)<\mathrm{\infty }\}\mathrm{.}$ For $x\in E_0$, define the coordinatewise barycenter $b_{\delta }(x)={({\int }_{\mathsf{D}}{a}_j\delta (x,da))}_{j=1}^d\mathrm{.}$ Assume that for every $\theta \in \mathrm{\Theta }$, $P_{\theta }(\{x\in E_0:b_{\delta }(x)\in \mathsf{D}\})=1.$ Then there is a measurable deterministic rule $d$ such that $R(\theta ,d)\le R(\theta ,\delta )(\theta \in \mathrm{\Theta })\mathrm{.}$ On $E_0$, finite first moment in the Euclidean norm of The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$ makes each coordinate integral finite. No claim is made for nonconvex action sets or when the parameterwise almost-sure barycenter condition fails.