Let $(E,\mathscr E)$ be a measurable space and $P,\tilde P$ be two stochastic kernels on that space. I wonder how the induced measures $\mathsf P_x$ and $\tilde{\mathsf P}_x$ differ on the space of finite trajectories $(\Omega_n,\mathscr F_n)$ and on the space of infinite trajectories $(\Omega,\mathscr F)$. I've already asked the finite-trajectories part of the question here and get two answers proving that $$ \sup\limits_{x\in E}\sup\limits_{C\in \mathscr F_n}|\tilde{\mathsf P}_x(C) - \mathsf P_x(C)| \leq n\|\tilde P - P\|. \tag{1} $$

Unfortunately, I didn't see in replies any canonical references on $(1)$ so I wonder if somebody has seen anything that looks like that result - or anything on the topic I've described.

I also wonder, when the rhs in $(1)$ can be bounded if we take the supremum over $C\in \mathscr F$, since clearly $(1)$ does not help in that case. I know that in general there are no trivial bounds since: there exists the example of the kernel $P$ and the pair $x,C$ such that $\mathsf P_x(C) = 1$ but for any $\delta>0$ there is $\tilde P^\delta$ such that $\|\tilde P^\delta - P\|\leq \delta$ but $\tilde{\mathsf P}_y(C) = 0$ for all $y$. On the other hand, I thought that some structural assumptions on $P$ may help: e.g. uniqueness of the invariant distribution and convergence of $P^n$ to this distribution. Again, I would be highly interested in references on the topic of how $(1)$ can be extended from $\mathscr F_n$ to $\mathscr F$, if there are any.

To be precise, I give here the formal statement of the problem.

Let $(E,\mathscr E)$ be a measurable space and for each $n\geq 0$ define $(\Omega_n,\mathscr F_n) = (E^{n+1},\mathscr E^{\otimes(n+1)})$ to be a measuruable product space. In the same fashion, let $\Omega = E^{\mathbb N_0}$ and $\mathscr F$ be its product $\sigma$-algebra.

Let $P$ be a stochastic kernel on $(E,\mathscr E)$, i.e. $P(x,\cdot)$ is a probability measure for each $x\in E$ and $P(\cdot,A)$ is a measurable function for each $A\in \mathscr E$. There is the unique family of measures $(\mathsf P_x)_{x\in E}$ on the space $(\Omega,\mathscr F)$ such that $$ \mathsf P_x(A_0\times A_1\times\dots \times A_n) = 1_{A_0}(x)\int\limits_{A_1}\dots\int\limits_{A_n}P(x_{n-1},\mathrm dx_n)\dots P(x,\mathrm dx_1) \tag{2} $$ where $n\geq0$ is any and $A_i\in \mathscr E$. Moreover, for each $n\geq 0$ let us denote $(\mathsf P^n_x)_{x\in E}$ to be the family of measures on the space $(\Omega_n,\mathscr F_n)$ which is defined uniquely by $(2)$.

Finally, the norm $\|\tilde P - P\|$ is defined by $$ \|\tilde P - P\| = \sup\limits_{f\in \mathrm b\mathscr E}\frac{\|\tilde Pf - Pf\|}{\|f\|} $$ where $\mathrm b\mathscr E$ is a Banach space of real-value bounded measurable functions with a norm $\|f\| = \sup\limits_{x\in E}|f(x)|$ and $$ Pf(x) = \int\limits_E f(y)P(x,\mathrm dy). $$