Finite-budget stopped-kernel measurability and event normalizer

For $m=0$, the three status probabilities are the indicators of the disjoint partition $T,R,S$; they are measurable and sum to one. Thus ${\kappa }_0$ is a probability kernel.

At $m=0$ the hit status occurs exactly when $X_0\in T$. For $m+1$, points in $T$ or $R$ stop immediately, while from $S$ the first $M$-transition is followed by the budget-$m$ rule. The finite-prefix law [F3] and induction therefore identify ${\kappa }_m(x,\{\mathtt{h}\mathtt{i}\mathtt{t}\})$ with $P_x({\tau }_T<{\tau }_R,{\tau }_T\le m)$; integrating against $\mu$ gives $p_m$.

Suppose ${\kappa }_m$ is a probability kernel. For each $c\in C$, the function $y\mapsto {\kappa }_m(y,\{c\})$ is measurable. By [F2], its $M(x,\cdot )$-integral is measurable in $x$. On $T$ and $R$ the next status is the corresponding point mass; on $S$ the three integrals sum to $\int 1M(x,dy)=1$. Hence ${\kappa }_{m+1}$ is a probability kernel, and induction proves the first claim.

In the one-step Feynman–Kac model, the product of potentials is ${\mathbf{1}}_{\{\mathtt{h}\mathtt{i}\mathtt{t}\}}$; [F4] therefore gives normalizer $\int {\mathbf{1}}_{\{\mathtt{h}\mathtt{i}\mathtt{t}\}}d(\mu {\kappa }_m)=p_m$. The success events by budgets $m$ increase to $\{{\tau }_T<{\tau }_R\}$; countable additivity gives the limit identity.