Let $A$ be a finite set, $\Omega = A^N$, with $N \in \mathbb{N}$ or $N = \infty$ and let $\mathcal{B}(\Omega)$ be a $\sigma$-algebra of $\Omega$. Consider $\omega \in \Omega$ such that $\omega = (x_1, x_2, \ldots ,x_N)$ and a sequence of random variables $(X_t)_{t=1,\ldots, N}$ such that $X_t(\omega) = x_t$. Assume that $(X_t)_{t=1,\ldots, N}$ is a homogeneous Markov chain with transition probability $P$ and initial probability $\mu$ such that $\mathbb{P}(X_1 = x_1, \ldots, X_k = x_k) = \mu(x_1)P(x_2|x_1)\ldots P(x_k|x_{k-1})$.

In the context of Markov decision processes, I found the following claim while reading [1]: let $R: \Omega \to \mathbb{R}$ and consider the following quantity:

$$ \mathbb{E}[R] = \sum_{\omega = (x_1, \ldots, x_N) \in \Omega} R(\omega) \mathbb{P}(X_1 = x_1, \ldots, X_N = x_N) $$ when $N$ is finite. When $N$ is infinite, the author claims that we can replace the sum above by an integral.

My question is: with respect to which measure would this integral be defined? And how is this measure related to the transition probability? I would like to understand precisely the meaning of

$$ \int_{\Omega} R(\omega) \mathbb{P}(\mathrm{d}\omega) $$ in this setting.

The probability of any trajectory is zero when $N = \infty$, so I thought about using a mapping $f: \Omega \to [0,1]$ such that $f(\omega) = \sum_{k \geq 1} x_k |A|^{-k}$, where $|A|$ is the number of elements in $A$ (that is, consider $\omega$ to be the representation in base $|A|$ of a real number between 0 and 1) and identifying a measure in $([0,1],\mathcal{B}([0,1))$ to a measure in $(\Omega,\mathcal{B}(\Omega))$. However, I do not know how to link this measure in $[0,1]$ to the transition probability of the Markov chain and whether this is a good approach.

I would also like to know how this generalizes when $A = \mathbb{R}$ instead of being finite.

Thank you for your help!

[1] Markov Decision Processes: Discrete Stochastic Dynamic Programming, by Martin L. Puterman, (section 2.1.6).

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