Let $P$ be a Feller transition probability on an unbounded space $X$, if there exists $x\in X$ such that the sequence $\{P^n(x, \cdot)\}_{n\ge 0}$ is tight, then show that there is an invariant probability measure for $P$.
The original Theorem assumes compactness and here in page-30, 1, he does not assume. could anyone tell me how then he is proving the result? I do not understand what would be wrong if space has not compactness, or completeness or boundedness or unboundedness is needed to prove the theorem. Thanks for helping.
Also, in the proof, the construction of probability measures $Q^n$ is not intuitive/natural to me.
Consider $X = \mathbb{Z}$ and $P(x, \cdot) = \delta_{x+1}(\cdot)$, i.e., $P$ defines a deterministic Markov chain which just moves one step to the right, so that $P^n(x, x+n) = 1$. This sequence is not tight and the chain has no stationary distribution.
The intuition for defining the measures $Q^n$ can be obtained by considering the Markov chain on two states, $X = \{0, 1\}$, with $P(0, 1) = P(1, 0) = 1$. You will see that $Q^n$ converges to $(1/2, 1/2)$, which is the correct stationary distribution.