In General Theory of Markov Processes, a sub-Markov semigroup on a measurable space $(E,\mathcal E)$ is extended to larger space containing an isolated point $\Delta\not\in E$ in the following way:
Why is this well-defined? Unless I'm missing something, $\tilde P_t(x,B)$ is not defined if $B$ is a proper superset of $\{\Delta\}$.
It's implicit that $\tilde P_t(x,\cdot)$ is a probability measure on $E_\Delta$. Thus, if $B=A\cup\{\Delta\}$ with $A\subset E$ then for $x\in E$, $$ \tilde P_t(x,B)=\tilde P_t(x,\Delta)+\tilde P_t(x,A)=[1-P_t(x,E)]+P_t(x,A). $$