Let $\{N(t),t\geq 0\}$ be a homogeneous Poisson process with rate $\lambda$. $N(t)$ is defined to be the number of events in $(0,t]$. Since $N(0)=0$ by definition, we can conclude that $P\{N(t)=i\}=P\{i \text{ events in }[0,t]\}$. Then, $N(t+s)-N(s)$ is the number of events in $(s,s+t]$, which is a Poisson random variable with parameter $\lambda t$. Since $P\{N(t+h)-N(t)=1\}=\lambda h+o(h)$ by definition, my intuition says that $$P\{N(t+s)-N(s)=i\}=P\{ i \text{ events in } [s,s+t]\}=P\{ i \text{ events in } [s,s+t)\}=P\{ i \text{ events in } (s,s+t)\}. $$
Question 1: Is my above intuition correct?
Question 2: If it is correct, how can we prove this rigorously? Can you suggest a related reference?
Any help will be really appreciated. Thanks in advance.
Your intuition is correct.
By construction, the process $(N_t)_{t \ge 0}$ is non-decreasing, right continuous. Moreover, for every $t>0$, the left limit $N_{t-}$ is the number of events in $(0,t)$.
The key point is that for each fixed time $t>0$, $N_t-N_{t-}$ is $0$ almost surely. Indeed, as $s \to t-$, $N_t-N_s \to N_t-N_{t-}$, and this almost sure convergence yields a convergence of distribution. Hence the law of $N_t-N_{t-}$ is the limit of Poisson($t-s$) as $s \to t-$, namely $\delta_0$.