Approximate-DP kernels admit a nearby pure-DP kernel on standard-Borel targets

Let $M(h,A)=(K(h,A)+L(h,A))\mathrm{/}2$, which is a probability kernel by [F2]. For every $h$, $M(h,A)=0$ implies $K(h,A)=L(h,A)=0$, so both sections are absolutely continuous with respect to $M(h,\cdot )$. Apply [F3] twice to obtain jointly measurable densities. Replace any value $+\mathrm{\infty }$ by $0$ in each density; this Borel operation preserves measurability by [F8] and changes no section measure on the common conull representation set, since each density has integral one. Call the resulting finite-valued functions $p(h,y)$ and $q(h,y)$. Let $\mathcal{C}$ be the countable generating algebra in [F4], containing $Y$. Define $G$ to be the set of $h$ for which, for every $C\in \mathcal{C}$, both $K(h,C)={\int }_{C}p(h,y)M(h,dy)$ and $L(h,C)={\int }_{C}q(h,y)M(h,dy)$ hold, and both DP inequalities hold for $C$. By [F2], [F5], and countability of $\mathcal{C}$, $G$ is measurable. It has full $\lambda$-measure by the joint representation property and the hypothesis. For $h\in G$, [F6] and [F7] make the density integrals finite measures; they agree with $K(h,\cdot )$ and $L(h,\cdot )$ on $\mathcal{C}$, so [F4] gives the density representations on every measurable event.

For fixed $h\in G$, put $\rho =K(h,\cdot )+L(h,\cdot )$, a finite measure. Let ${\mathcal{E}}_{\rho }$ be the measurable sets approximable in $\rho$-measure by members of $\mathcal{C}$. Complements preserve symmetric-difference measure, and $(A\cup B)\mathrm{△}(C\cup D)\subseteq (A\mathrm{△}C)\cup (B\mathrm{△}D)$ proves closure under finite unions. If $A_n\in {\mathcal{E}}_{\rho }$ and $A={\bigcup }_n{A}_n$, put $U_N={\bigcup }_{n\le N}{A}_n$ and $D_1=U_1$, $D_N=U_N\setminus U_{N-1}$ for $N>1$. The $D_N$ are disjoint and $A={\bigcup }_N{D}_N$, so countable additivity and finiteness of $\rho (A)$ give $\rho (A\setminus U_N)={\sum }_{j>N}\rho (D_j)\to 0$. Approximate each of the first $N$ sets closely enough that their finite union has arbitrarily small symmetric-difference measure from $U_N$. Thus ${\mathcal{E}}_{\rho }$ is a sigma-algebra containing $\mathcal{C}$, so it is the full target sigma-algebra. For any measurable $B$, choose $C_n\in \mathcal{C}$ with $\rho (B\mathrm{△}{C}_n)\to 0$. Then both $K(h,C_n)\to K(h,B)$ and $L(h,C_n)\to L(h,B)$, so the two inequalities on $\mathcal{C}$ pass to every measurable $B$.

Write $a=\exp (\epsilon )\ge 1$ and $c=\exp (-\epsilon )=1\mathrm{/}a>0$ by [F9]. On $G$, set $p_0=\min \{\max \{p,cq\},aq\}$, $\alpha =\int (p-aq{)}_{+}dM$, and $\beta =\int (cq-p{)}_{+}dM$. These quantities are finite and measurable in $h$ by [F5] and [F8]. The sets $A_h=\{y:p(h,y)>aq(h,y)\}$ and $B_h=\{y:p(h,y)<cq(h,y)\}$ are measurable by [F7]. Step 2.1 and the density identities give $\alpha =K(h,A_h)-aL(h,A_h)\le \delta$ and $a\beta =L(h,B_h)-aK(h,B_h)\le \delta$. Also $p_0=p-(p-aq{)}_{+}+(cq-p{)}_{+}$, so $m:=\int p_{0}dM=1-\alpha +\beta$. Hence $\max \{\alpha ,\beta \}\le \delta$.

Define a corrected density on $G$ as follows. If $m<1$, let $t_{+}=(1-m)\mathrm{/}(a-m)$ and $r=p_0+t_{+}(aq-p_0)$. Its denominator is positive and $0\le t_{+}\le 1$ because $a\ge 1$, so $p_0\le r\le aq$ and $\int rdM=m+t_{+}(a-m)=1$. If $m>1$, let $t_{-}=(m-1)\mathrm{/}(m-c)$ and $r=p_0-t_{-}(p_0-cq)$. Here $m-c>0$ and $0\le t_{-}\le 1$ because $c\le 1$, so $cq\le r\le p_0$ and $\int rdM=m-t_{-}(m-c)=1$. If $m=1$, set $r=p_0$. These formulas are product-measurable by [F8], and in every case $cq\le r\le aq$ and $\int rdM=1$. Set $r=0$ off $G$ and define $K^{\ast }(h,A)={\int }_{A}r(h,y)M(h,dy)+{\mathbf{1}}_{G^c}(h)L(h,A)$. By [F5], this is measurable in $h$ for each $A$; for each $h\in G$ it is a probability measure by [F6] and the normalization, and for $h\mathrm{\notin }G$ it equals the probability measure $L(h,\cdot )$. Thus $K^{\ast }$ is a probability kernel. For $h\in G$, integrate $cq\le r\le aq$ and use the density representation of L to obtain the claimed two-sided pure-DP inequalities.

For $h\in G$, clipping changes $p$ by total integral $\int \mathrm{\mid }p-p_0\mathrm{\mid }dM=\alpha +\beta$, while correction moves $p_0$ in one direction and changes its integral by $\mathrm{\mid }m-1\mathrm{\mid }=\mathrm{\mid }\alpha -\beta \mathrm{\mid }$. Thus $\int \mathrm{\mid }p-r\mathrm{\mid }dM\le \alpha +\beta +\mathrm{\mid }\alpha -\beta \mathrm{\mid }=2\max \{\alpha ,\beta \}\le 2\delta$. For probability densities with equal integral, ${\sup }_A\mathrm{\mid }{\int }_A(p-r)dM\mathrm{\mid }=\frac{1}{2}\int \mathrm{\mid }p-r\mathrm{\mid }dM$: every event integral is bounded by the positive-part integral and $\{p\ge r\}$ attains it. Hence $\mathrm{TV}(K(h,\cdot ),K^{\ast }(h,\cdot ))\le \delta$ on $G$. This TV function is measurable: applying the approximation argument of step 2.1 to $K+K^{\ast }$ shows its supremum may be taken over $\mathcal{C}$, whose kernel evaluations are measurable. Since $G$ is conull, integration proves the asserted bound.