Could anyone tell me what is $P(x,dy)$ and $P^{n+1}(x,dy)$ means here in 1p- 30, in the proof of thm 4.17? And, why $\phi$ was taken bounded by $1$? Are all $P^k(x,A)$, $P$, $Q^n$ probability measure on $(X, B(X))$? Thanks for helping.
2026-03-26 14:18:44.1774534724
understanding Krylov-Bogolubov Theorem
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For each fixed $x$ $P(x,A)$ is a probability measure and the integral of $f$ w.r.t. this measure is written as $\int f(y)P(x,dy)$. Similar interpretation holds for $\int f(y)P^{n}(x,dy)$.