My question is related to the renewal process, as defined in this document.
Renewal process is an arrival process in which the interarrival intervals are positive, independent and identically distributed random variables denoted $\{X_n; n \ge 1\}$.
$\{N(t); t\ge 0\}$ is a renewal process with inter-renewal random variables $\{X_n; n \ge 1\}$. $N(t)$ is the number of arrivals to a system in the interval $(0,t]$.
See page 3 for lemma 3.1. The first sentence of the proof says:
Note that for each sample point $\omega \in \Omega$, $N(t,\omega)$ is a nondecreasing function of $t$ and thus either has a finite limit or an infinite limit.
What is the sample space $\Omega$ of a renewal process?
My guess: Let's denote the observed value of r.v. $X_i$ by $x_i$.
What is a random variable? It's a function that assigns a value to every outcome from the sample space $\Omega$. If we have the sequence of random variables (that sequence can be infinite), $\{X_n\}$, then we need to define what $\Omega$ is.
I'd say $\Omega$ has to be the collection of all possible infinite sequences of the form $\{x_1, x_2, x_3, ...\}$. Then essentially the random variable $X_i$ is equal to the value on the $i$th position of particular outcome from the sample space $\Omega$. A single outcome is an infinite sequence, because the renewal process consists of an infinite sequence of random variables $X_n$.
Is there anything that should be added or corrected in my definition of the sample space here?
Your definition is similar to the usual one, except that you don't want to have separate points $\omega\in\Omega$ for the finite sequences. Just treat them as prefixes of infinite sequences. The $X_i=X_i(\omega)$ you mention are usually called the coordinate functions. So the probability that the first arrival is less than 3 is the measure on $\Omega$ of $\{\omega:X_1(\omega)<3\}$. The measure on $\Omega$ is determined by the distribution of the $X_i$. You might look up Kolmogorov's extension theorem for details. The process $N(t,\omega)$, although it has a continuous time parameter, is determined by the $X_i$, so you can define it on the same probability space, i.e., $N(t,\omega)$ is a random variable on the sample space of sequences, just like the coordinate functions. An event about $N(t)$ is an event about the $X_i$.