Let $T_n$ be an independent sequence of exponential random variables with rate parameter $\lambda$ and $W_n = T_1 + \cdots +T_n$.
Define the random variable $N_t(\omega) = \sum_{j = 1}^{\infty} 1_{\{W_j (\omega) \le t\}}$ and $N_0 = 0$ almost surely, which counts the number of events occurring in $[0,t]$
I was able to prove that for fixed $0 \le s < t$: $N_s \le N_t$ almost surely but I can't find a way to prove that:
$P[N_s \le N_t,\text{ } \text{ }\forall \text{ }0 \le s < t] = 1$
this is because the set: $$\{ \omega \in \Omega \text{ } |\text{ } N_s(\omega) \le N_t(\omega),\text{ } \text{ }\forall \text{ }0 \le s < t\} = \bigcap_{0 \le s < t} \{ \omega \in \Omega \text{ } | \text{ }N_s(\omega) \le N_t(\omega) \}$$
is an uncountable intersections of events which needs not to be measurable.
Likewise I can't find a way to prove almost sure piecewise constant trajectories without any extra assumptions.
I have two questions regarding this issue:
Without any extra assumptions is there an example of a process such that: $N_t(\omega) = \sum_{j = 1}^{\infty} 1_{\{W_j (\omega) \le t\}}$ but does not have non-decreasing or piecewise constant trajectories almost surely?.
What are the extra assumptions that we normally need to have in order to prove non-decreasing and piecewise constant trajectories almost surely? I have seen in some textbooks that we impose:
a) Right continuous trajectories and non-decreasing almost surely. or
b) Continuity in probability of the trajectories. or
c) Cadlag process.
Which of the above is the most standard extra assumptions such that the process works fine? or are they all equivalent?
I would really appreciate any comment regarding this issue.