Killed and absorbed kernels are probability kernels

Fix $x\in E$. If $x\in D$, the absorbed section in [F1] is ${\delta }_x$; if [F1, F2] $x\in D^c$, it is $K(x,\cdot )$. Hence every section is countably additive, has empty-set mass zero and total mass one. For fixed $A\in \mathcal{E}$, its evaluation is $1_D(x)1_A(x)+1_{D^c}(x)K(x,A),$ a measurable function by [F2]. Thus $K_D^{\mathrm{a}\mathrm{b}\mathrm{s}}$ is a probability kernel.

Fix $x\in D$. The killed section is the restriction [F1, F2] $A\mapsto K(x,A\cap D)$ on $E$, plus an atom of mass $K(x,D^c)$ at $\mathrm{\Delta }$. It is countably additive and its total mass is $K(x,D)+K(x,D^c)=1$. If $x\in D^c\cup \{\mathrm{\Delta }\}$, the section is ${\delta }_{\mathrm{\Delta }}$. Thus all killed sections are probability measures, including $D=\mathrm{\varnothing }$ and $D=E$.

Fix $B\in {\mathcal{E}}_{\mathrm{\Delta }}$, put $A=B\cap E$, and let [F1, F2, step 1.2] $\epsilon =1_B(\mathrm{\Delta })$. On $E$ the killed evaluation is $1_D(x)\{K(x,A\cap D)+\epsilon K(x,D^c)\}+1_{D^c}(x)\epsilon ,$ which is $\mathcal{E}$-measurable by [F2]. Its value at the measurable singleton $\{\mathrm{\Delta }\}$ is $\epsilon$, so the full evaluation is ${\mathcal{E}}_{\mathrm{\Delta }}$-measurable. Therefore $K_D^{\mathrm{k}\mathrm{i}\mathrm{l}\mathrm{l}}$ is a probability kernel. The empty and full target sets give respectively zero and one in every case. No choice principle is used.