After equation (1) at page 3 of this paper it is claimed that the transition kernel of a Markov chain generated by the Metropolis-Hastings algorithm is of the form $$P(x,A)=\varrho(x)\tilde P(x,A)+(1-\varrho(x))1_A(x)\;,\;\;\;(x,A)\in E\times\mathcal E\tag1,$$ where $\tilde P$ is a Markov kernel on a measurable space $(E,\mathcal E)$ and $\varrho:E\to(0,1]$ is $\mathcal E$-measurable.
How do we see this? The transition kernel $\kappa$ of the Metropolis-Hastings algorithm with proposal kernl $Q$ and acceptance function $\alpha$ is given by $$\kappa(x,B)=\int_BQ(x,{\rm d}y)\alpha(x,y)+r(x)1_B(x)\;,\;\;\;(x,B)\in E\times\mathcal E,$$ where $$r(x):=1-\int Q(x,{\rm d}y)\alpha(x,y)\;\;\;\text{for }x\in E.$$
I don't see how $\tilde P$ should be defined to match the form.