A stable two-point potential gives a Lipschitz posterior

For any $a\in \mathbb{R}$, the intervals $(x-\mathrm{\mid }x-a\mathrm{\mid }\mathrm{/}2,x+\mathrm{\mid }x-a\mathrm{\mid }\mathrm{/}2)$, indexed by $x\mathrm{\ne }a$, cover $\mathbb{R}\setminus \{a\}$ and omit $a$; hence $\mathbb{R}\setminus \{a\}$ is open and every singleton is closed. Therefore $D$ is Borel as a finite union of closed sets, and it is bounded because $\mathrm{\mid }u\mathrm{\mid }=1$ on $D$. The open intervals $(1\mathrm{/}2,3\mathrm{/}2)$ and $(-3\mathrm{/}2,-1\mathrm{/}2)$ intersect $D$ in $\{1\}$ and $\{-1\}$, respectively, so every singleton is relatively open and $D$ has the discrete relative topology. The displayed finite atomic measure is countably additive and its two atom masses sum to one, so it is a Borel probability with ${\mu }_0(D)=1$. Thus $D$ itself is a bounded Borel subset of positive prior mass, as required by the theorem. The real line is a separable Banach space. By [F4], for any open $O\subseteq \mathbb{R}$, ${\mathrm{\Phi }}^{-1}(O)$ intersects each open slice $\{u\}\times \mathbb{R}$ in an open subset of that slice, hence open in $D\times \mathbb{R}$; taking the union over $u\in D$ proves joint continuity. The potential is nonnegative. For $\mathrm{\mid }y\mathrm{\mid },\mathrm{\mid }{y}^{\mathrm{\prime }}\mathrm{\mid }\le R$ and $u\in D$, $\mathrm{\mid }\mathrm{\Phi }(u;y)-\mathrm{\Phi }(u;y^{\mathrm{\prime }})\mathrm{\mid }=\frac{1}{2}\mathrm{\mid }y-y^{\mathrm{\prime }}\mathrm{\mid }\mathrm{\mid }y+y^{\mathrm{\prime }}-2u\mathrm{\mid }\le (R+\mathrm{\mid }u\mathrm{\mid })\mathrm{\mid }y-y^{\mathrm{\prime }}\mathrm{\mid }\mathrm{.}$ Here $\mathrm{\mid }y+y^{\mathrm{\prime }}-2u\mathrm{\mid }\le \mathrm{\mid }y\mathrm{\mid }+\mathrm{\mid }{y}^{\mathrm{\prime }}\mathrm{\mid }+2\mathrm{\mid }u\mathrm{\mid }\le 2R+2\mathrm{\mid }u\mathrm{\mid }$ by the triangle inequality. Thus choose $M_1(r,s)=0$ and $M_2(r,s)=r+s$, which are Borel and nondecreasing in each argument. Then ${\int }_D{e}^{M_1(R,\mathrm{\mid }u\mathrm{\mid })}(1+M_2(R,\mathrm{\mid }u\mathrm{\mid }{)}^2)d{\mu }_0(u)=1+(R+1{)}^2<\mathrm{\infty },$ so [F2] applies and gives the stated Hellinger Lipschitz stability on $[-R,R]$.

By [F1], the evidence is the displayed two-term sum and is positive. The posterior mass ratio is $\frac{{\mathrm{\Pi }}^y(\{1\})}{{\mathrm{\Pi }}^y(\{-1\})}=\frac{e^{-(y-1{)}^2\mathrm{/}2}}{e^{-(y+1{)}^2\mathrm{/}2}}=e^{2y};$ the two masses sum to one, so ${\mathrm{\Pi }}^y(\{1\})=e^{2y}\mathrm{/}(1+e^{2y})$. The verified evidence and the explicit formula determine the whole two-point posterior.