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 Apr 30 awarded Nice Question Jul 2 awarded Curious Jun 25 awarded Yearling Oct 26 awarded Autobiographer Oct 26 comment How to prove that $P^n(x,\cdot)\stackrel{w}{\rightarrow} \pi$ for all $x$, implies the equicontinuity of $\{P^n f\}_n$ on compact sets? I mean the Arzela-Ascoli theorem says that if $(f_n)$ is a sequence of continuos functions on a compact set then each subsequence of $(f_n)$ has a unformly convergent subsequence iff $\{f_n\}$ is equicontinous and pointwise bounded. So, you are right, if $(f_n)$ is pointwise convergent it equivalent to the uniform convergence of $(f_n)$. Oct 26 accepted How to prove that $P^n(x,\cdot)\stackrel{w}{\rightarrow} \pi$ for all $x$, implies the equicontinuity of $\{P^n f\}_n$ on compact sets? Oct 26 comment How to prove that $P^n(x,\cdot)\stackrel{w}{\rightarrow} \pi$ for all $x$, implies the equicontinuity of $\{P^n f\}_n$ on compact sets? Under the strong Feller property, you have shown that $(P^n f)_n$ is uniformly convergent on compacts sets, which is even stronger that we need. In general, I have tried something similar to this. Namely I have tried to show that a the subsequence of $(|P^n f - \pi f|)_n$ is decreasing on the set $K_0=\{x_n: n\in\mathbb{N}\} \cup \{x_0\}$, where $x_0\in K$ and $x_n$ is an arbitrary sequence of the set $K$ convergent to $x_0$. The we would have the uniform convergence on the set $K_0$ by the Dini Theorem, and therefore, on the set $K$. Oct 26 comment How to prove that $P^n(x,\cdot)\stackrel{w}{\rightarrow} \pi$ for all $x$, implies the equicontinuity of $\{P^n f\}_n$ on compact sets? The Dini Theorem was the first thing I have tried to use in poving this. Oct 25 comment How to prove that $P^n(x,\cdot)\stackrel{w}{\rightarrow} \pi$ for all $x$, implies the equicontinuity of $\{P^n f\}_n$ on compact sets? The authors write "Since the limit in (6.20) is continuous (and in fact constant) it follows from Ascoli’s Theorem (...)". This sentence sounds a little strange for me. Oct 25 comment How to prove that $P^n(x,\cdot)\stackrel{w}{\rightarrow} \pi$ for all $x$, implies the equicontinuity of $\{P^n f\}_n$ on compact sets? Thanks for the reply. Now, at least, I know it is not obvious. Maybe it should be assumed that the weak convergence of $P^n(x,\cdot)$ is uniform with respect to $x$ on compact sets. Oct 25 revised How to prove that $P^n(x,\cdot)\stackrel{w}{\rightarrow} \pi$ for all $x$, implies the equicontinuity of $\{P^n f\}_n$ on compact sets? added 1 characters in body Oct 25 revised How to prove that $P^n(x,\cdot)\stackrel{w}{\rightarrow} \pi$ for all $x$, implies the equicontinuity of $\{P^n f\}_n$ on compact sets? deleted 8 characters in body Oct 25 revised How to prove that $P^n(x,\cdot)\stackrel{w}{\rightarrow} \pi$ for all $x$, implies the equicontinuity of $\{P^n f\}_n$ on compact sets? added 59 characters in body Oct 25 asked How to prove that $P^n(x,\cdot)\stackrel{w}{\rightarrow} \pi$ for all $x$, implies the equicontinuity of $\{P^n f\}_n$ on compact sets? Jun 25 awarded Yearling Feb 2 accepted Is the function $B(x,y,\cdot)=P(x,\cdot)\otimes P(y,\cdot)$ measurable? Feb 2 comment Is the function $B(x,y,\cdot)=P(x,\cdot)\otimes P(y,\cdot)$ measurable? Thanks! I can not believe it could be so easy:) Feb 1 revised Is the function $B(x,y,\cdot)=P(x,\cdot)\otimes P(y,\cdot)$ measurable? deleted 3 characters in body Feb 1 asked Is the function $B(x,y,\cdot)=P(x,\cdot)\otimes P(y,\cdot)$ measurable? Dec 23 comment How to prove that $\lim_{n\to\infty} n^{-1} \ln (\|A^n\|)=\ln(\rho(A))$, where $A$ is a matrix. Yes, you are right, a was looking everywhere but not in wikipedia :)