Attained entropy-maximizing mixtures on compact phase domains

The map $h$ is continuous, so [L2] makes $H$ compact. For every Borel probability $P$ on $K$, each coordinate of $h$ is bounded and Borel. Compactness and [L1] give boundedness. If $\mathrm{\mid }f\mathrm{\mid }\le M$ for one such coordinate, [F3] gives ${\int }_K\mathrm{\mid }f\mathrm{\mid }dP\le MP(K)=M$, so $f$ is integrable.

Every point of $C$ is a finite convex combination of points $h(x_i)$. If such a representation has $N>d+2$, the linear map $T:{\mathbb{R}}^N\to {\mathbb{R}}^{d+2}$ defined by $T(c)=({\sum }_i{c}_{i}h(x_i),{\sum }_i{c}_i)$ has a nonzero kernel vector $c$ by [L3], because the coordinate bases give domain dimension $N$ and codomain dimension $d+2$. Thus ${\sum }_i{c}_i=0$ and ${\sum }_i{c}_{i}h(x_i)=0$, so $c$ has both positive and negative coordinates. After dropping zero weights from the representation, set $t={\min }_{c_i>0}{\theta }_i\mathrm{/}{c}_i$ and replace each weight ${\theta }_i$ by ${\theta }_i-t{c}_i$. The new weights remain nonnegative, still sum to one, represent the same point, and at least one is zero. Repeating reduces every combination to at most $d+2$ terms. Hence $C$ is the image of $K^{d+2}\times {\mathrm{\Delta }}_{d+2}$ under $\mathrm{\Theta }((x_i),({\theta }_i))={\sum }_{i=1}^{d+2}{\theta }_{i}h(x_i)$, where ${\mathrm{\Delta }}_{d+2}:=\{({\theta }_i):{\theta }_i\ge 0,{\sum }_i{\theta }_i=1\}$; shorter combinations are padded with zero weights and any fixed point of nonempty $K$. Each factor $K$ is closed and bounded by [L2], and ${\mathrm{\Delta }}_{d+2}$ is closed by continuity of the finite coordinate sum and its coordinate inequalities; it is bounded since $0\le {\theta }_i\le 1$. Their product is therefore closed and bounded in ${\mathbb{R}}^{(d+1)(d+2)}$, so compact by [L2]. To check $\mathrm{\Theta }$ is continuous on this compact parameter set, fix a parameter $((x_i),({\theta }_i))$. Each coordinate $h_j$ is bounded by some $M$ on $K$ and is continuous at each of the finitely many $x_i$. For any $ϵ>0$, choose neighborhoods of the $x_i$ making $\mathrm{\mid }{h}_j(x_i^{\mathrm{\prime }})-h_j(x_i)\mathrm{\mid }<ϵ\mathrm{/}(2N)$ for every output coordinate $j$, and require $\mathrm{\mid }{\theta }_i^{\mathrm{\prime }}-{\theta }_i\mathrm{\mid }<ϵ\mathrm{/}(2N(M+1))$. Since $0\le {\theta }_i^{\mathrm{\prime }}\le 1$ and $\mathrm{\mid }{h}_j(x_i)\mathrm{\mid }\le M$, each product difference $\mathrm{\mid }{\theta }_i^{\mathrm{\prime }}{h}_j(x_i^{\mathrm{\prime }})-{\theta }_i{h}_j(x_i)\mathrm{\mid }<ϵ\mathrm{/}N$; summing over $i$ makes each output-coordinate difference smaller than $ϵ$. Thus $\mathrm{\Theta }$ is continuous. The continuous-image clause of [L2] makes its image $C$ compact, and [L2] also makes $C$ closed.

Fix any Borel probability $P$ on $K$ and let $y(P):=({\int }_{K}xdP,{\int }_{K}sdP)={\int }_{K}hdP$, with coordinatewise integration. For $ϵ>0$, compactness of $H$ gives a finite cover of $H$ by open Euclidean balls of radius $ϵ$. Their preimages under $h$ are Borel; disjointifying them in a fixed finite order gives a finite Borel partition $(A_i)$ of $K$. For every nonempty part choose $z_i\in A_i$ and put ${\theta }_i=P(A_i)$. Then $y_{ϵ}:={\sum }_i{\theta }_{i}h(z_i)$ belongs to $C$. For $x\in A_i$, both $h(x)$ and $h(z_i)$ lie in the same radius-$ϵ$ ball, so each coordinate difference is at most $2ϵ$. The simple map equal to $h(z_i)$ on $A_i$ has integral $y_{ϵ}$ by the simple-integral definition. By linearity and the integral triangle inequality in [F3], each coordinate of $y(P)-y_{ϵ}$ has absolute value at most $2ϵ{\sum }_{i}P(A_i)=2ϵ$. Consequently $\mathrm{\parallel }y(P)-y_{ϵ}{\mathrm{\parallel }}_2\le 2ϵ\sqrt{d+1}$, so the sequence obtained with $ϵ=1\mathrm{/}n$ converges to $y(P)$. Since $C$ is closed by step 2.1, $y(P)\in C$.

Because $\bar{x}\in \mathrm{conv}K$, a finite convex combination of generators has barycenter $\bar{x}$; its corresponding combination of points $h(x)$ shows that $E_{\bar{x}}:=\{r\in \mathbb{R}:(\bar{x},r)\in C\}$ is nonempty. The set $E_{\bar{x}}$ is bounded because $C$ is bounded, and it is closed because $C$ is closed and $r\mapsto (\bar{x},r)$ is continuous. By [L2] it is compact in $\mathbb{R}$, so [L1] applied to the identity function yields a maximum $r_{\ast }\in E_{\bar{x}}$.

Represent $(\bar{x},r_{\ast })\in C$ by a finite convex combination ${\sum }_{i=1}^N{\theta }_i(x_i,s(x_i))$, dropping zero weights. If $N>d+1$, the linear map $R:{\mathbb{R}}^N\to {\mathbb{R}}^{d+1}$ given by $R(c)=({\sum }_i{c}_i{x}_i,{\sum }_i{c}_i)$ has a nonzero kernel vector by [L3], since its domain dimension is $N$ and its codomain dimension is $d+1$. Hence ${\sum }_i{c}_i=0$ and ${\sum }_i{c}_i{x}_i=0$, with both signs among the $c_i$. If ${\sum }_i{c}_{i}s(x_i)\mathrm{\ne }0$, choose the sign of $c$ so that this sum is positive. For sufficiently small $t>0$, the weights ${\theta }_i+t{c}_i$ are nonnegative, preserve total mass and barycenter, and increase the last coordinate above $r_{\ast }$, contradicting maximality in step 3.2. Thus ${\sum }_i{c}_{i}s(x_i)=0$. Now take $t={\min }_{c_i>0}{\theta }_i\mathrm{/}{c}_i$ and replace ${\theta }_i$ by ${\theta }_i-t{c}_i$. This removes at least one positive weight while preserving total mass, barycenter, and entropy value $r_{\ast }$. Repeating finitely many times leaves at most $d+1$ positive weights.

Let $P_{\ast }:={\sum }_i{\theta }_i{\delta }_{x_i}$ be the probability measure from step 4.1. Each ${\delta }_{x_i}$ is countably additive because among disjoint Borel sets at most one contains $x_i$, and ${\delta }_{x_i}(K)=1$; hence $P_{\ast }$ is a Borel probability measure. It has barycenter $\bar{x}$ and entropy $r_{\ast }$, so it is feasible by the model assumption. For any other feasible Borel probability $P$, step 3.1 gives $(\bar{x},\int sdP)\in C$, so $\int sdP\le r_{\ast }$. Thus $P_{\ast }$ attains the entropy maximum among all Borel mixtures and has at most $d+1$ atoms. No finite generator menu was assumed.