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Let $(S,\mathcal{S})$ be a measurable space (thought of as the state space of a Markov chain). In Durrett's probability, he defines a transition probability to be a function $p:(S,\mathcal{S}) \to \mathbb{R}$ such that

  1. For fixed $x \in S$, $p(x,\cdot)$ is a probability measure on $(S,\mathcal{S})$.
  2. For fixed $A \in \mathcal{S}$, $p(\cdot, A)$ is a measurable function.

Then he goes on to say that $X_n$ is called a Markov chain (wrt $\mathcal{F}_n$) with transition probability $p$ if $P(X_{n+1} \in B | \mathcal{F}_n) = p(X_n,B)$. But what does $p(X_n,B)$ mean? $X_n$ is a function $X_n: \Omega \to S$, not an element of $S$. Do we interpret $X_n$ as having a fixed value $X_n=s$ and then interpret $p(X_n,B)$ as being $p(s,B)$?

Another confusing bit of notation: he goes on to define the canonical sequence space $(S^\mathbb{N}, \mathcal{S}^\mathbb{N})$ and a probability measure $P_\mu$ on it via the Kolmogorov extension theorem. He says that $P_\mu(A) = \int P_x(A) d\mu(x)$, but he never defines what $P_x$ is.

Justin Furlotte
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  • I think you are meant to read previous chapters before reading that. In particular, he is using the notation in his chapter on martingales and regular conditional probabilities without restating them. – user10354138 Mar 10 '21 at 12:13
  • @user10354138 I did read the previous chapter on Martingales. Neither of these notations was defined there. My background on probability theory came from a course where we used Rosenthal, so I unfortunately didn't read the first three chapters of Durrett, but after a couple of hours of skimming them I don't see anywhere he defined this notations in the first three chapters either. – Justin Furlotte Mar 10 '21 at 18:04
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    Did you figure it out by any chance? I'm also struggling to understand some notations despite having gone through the first 4 chapters... – MoneyBall Mar 14 '21 at 08:39
  • @MoneyBall Sorry for the late reply - unfortunately I haven't figured out how to interpret $p(X_n,B)$ but I did figure out that $P_x(A)$ denotes "probability of $A$ given that the Markov chain starts at $x$", and $P_\mu(A)$ denotes "probability of $A$ given that the Markov chain has initial distribution $\mu$". – Justin Furlotte Mar 27 '21 at 00:36

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