Problem:
Let $N_1 , ...., N_n$ be independent Poisson processes on $[0,\infty)$ defined on the same probability space.
Show that $N_1 + N_2 + ... + N_n$ is a Poisson process and determine its mean value function.
Attempt:
I have shown all other properties, but I'm yet to show the increment property which states:
"There exists a non-decreasing right-continuous function $\mu : [0,\infty) \to [0,\infty)$ with $\mu(0)=0$ such that the increments $N(s,t]$ for $0<s<t<\infty$ have a Poisson distribution $Pois(\mu(s,t])$."
Notation: $f(s,t]=f(t)-f(s)$
If i had $L = N_1 + N_2$ then i think i would have to assume that i have the intervals:
$$[0,t_1] \text{ and } [t_1,t_2]$$
for $0=t_0 < t_1 <t_2 $
But what do I do now?
My idea was (for the $n=2$ case) to show that $$P(N(0,t_1]+N(t_1,t_2]=k + \ell)=e^{-(\mu(0,t_1]+\mu(t_1,t_2])}\frac{(\mu(0,t_1])^k (\mu(t_1,t_2])^\ell}{k ! \ell !}$$
But is this true? if not, please give me some hints...