ABC targets the density-defined summary posterior at zero tolerance

Fix $0<ϵ<{ϵ}_0$. The Euclidean distance is a metric by [F3], so its closed ball $B_{ϵ}(s_0)$ is closed and therefore Borel. Under [A1], [F3] makes it Lebesgue measurable. It contains the open cube of side $2ϵ\mathrm{/}\sqrt{d}$ centered at $s_0$ and is contained in the closed cube of side $2ϵ$ centered there: the first inclusion follows from $\mathrm{\parallel }x-s_0{\mathrm{\parallel }}_2^2<d(ϵ\mathrm{/}\sqrt{d}{)}^2={ϵ}^2$, and the second from $\mathrm{\mid }{x}_i-s_{0,i}\mathrm{\mid }\le ϵ$. The box-volume formula and monotonicity in [F3] give $0<(2ϵ\mathrm{/}\sqrt{d}{)}^d\le v_{ϵ}:={\lambda }_d(B_{ϵ}(s_0))\le (2ϵ{)}^d<\mathrm{\infty }\mathrm{.}$

Define $q_{ϵ}(\theta ):=\frac{1}{v_{ϵ}}{\int }_{B_{ϵ}(s_0)}{m}_{\theta }(s){\lambda }_d(ds)\mathrm{.}$ By step 1.1, $0<v_{ϵ}<\mathrm{\infty }$, so ${\mathrm{\Lambda }}_{ϵ}(\theta ,C):={\lambda }_d(C\cap B_{ϵ}(s_0))$ is a finite constant kernel from $\mathrm{\Theta }$ to $({\mathbb{R}}^d,\mathcal{L}({\mathbb{R}}^d))$. The stated joint measurability of $m$ and [F4] applied to ${\mathrm{\Lambda }}_{ϵ}$ make $q_{ϵ}$ measurable. The density identity and [F5] give, for every measurable $A\subseteq \mathrm{\Theta }$, $M(A\times B_{ϵ}(s_0))=v_{ϵ}{\int }_A{q}_{ϵ}(\theta )\mathrm{\Pi }(d\theta )\mathrm{.}$ Thus, whenever $z_{ϵ}:=\int q_{ϵ}d\mathrm{\Pi }>0$, the ABC target has density $q_{ϵ}\mathrm{/}{z}_{ϵ}$ relative to $\mathrm{\Pi }$. The envelope bounds $q_{ϵ}\le H$, so $z_{ϵ}<\mathrm{\infty }$.

Take an arbitrary sequence ${ϵ}_j>0$ with ${ϵ}_j\to 0$. Fix any $\theta$ in the full-measure set where $m_{\theta }$ is continuous at $s_0$ and bounded by $H(\theta )$. Given $\eta >0$, continuity supplies $\delta >0$ such that $\mathrm{\mid }{m}_{\theta }(s)-m_{\theta }(s_0)\mathrm{\mid }<\eta$ whenever $\mathrm{\parallel }s-s_0{\mathrm{\parallel }}_2<\delta$. For all large $j$, ${ϵ}_j<\delta$, and averaging over $B_{{ϵ}_j}(s_0)$ gives $\mathrm{\mid }{q}_{{ϵ}_j}(\theta )-m_{\theta }(s_0)\mathrm{\mid }\le \eta$. Thus pointwise convergence holds for $\mathrm{\Pi }$-almost every $\theta$. Also $\mathrm{\mid }{q}_{{ϵ}_j}-m_{\theta }(s_0)\mathrm{\mid }\le 2H$ and $\int Hd\mathrm{\Pi }<\mathrm{\infty }$, so [F6] yields ${\int }_{\mathrm{\Theta }}\mathrm{\mid }{q}_{{ϵ}_j}(\theta )-m_{\theta }(s_0)\mathrm{\mid }\mathrm{\Pi }(d\theta )⟶0.$ Since the sequence was arbitrary, [F10] gives this $L^1$ limit as $ϵ\downarrow 0$. In particular $z_{ϵ}\to z_0>0$, and therefore $z_{ϵ}>0$ for every sufficiently small positive $ϵ$.

For such $ϵ$, the normalized densities satisfy ${\int }_{\mathrm{\Theta }}\mid \frac{q_{ϵ}}{z_{ϵ}}-\frac{m_{\theta }(s_0)}{z_0}\mid d\mathrm{\Pi }\le \frac{1}{z_{ϵ}}{\int }_{\mathrm{\Theta }}\mathrm{\mid }{q}_{ϵ}-m_{\theta }(s_0)\mathrm{\mid }d\mathrm{\Pi }+\frac{\mathrm{\mid }{z}_{ϵ}-z_0\mathrm{\mid }}{z_{ϵ}}\mathrm{.}$ The right side tends to zero by step 3.1. The second density integrates to one by [F8] and $0<z_0<\mathrm{\infty }$; the first is the ABC target density by step 2.1.

For every measurable $A\subseteq \mathrm{\Theta }$, the absolute difference between the two probabilities of $A$ is at most the $L^1(\mathrm{\Pi })$ difference in step 4.1. Taking the supremum over $A$ and using [F7] proves $\mathrm{\parallel }{\mathrm{\Pi }}_{ϵ}(\cdot \mid s_0)-{\mathrm{\Pi }}_{s_0}^m{\mathrm{\parallel }}_{\mathrm{T}\mathrm{V}}\to 0$. The limit is the density-selected update; by [F9], no claim is made that every RCD version takes that value at a marginal-null $s_0$.