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What I have learned from my recent readings is that if $\{N_t,t\geq 0\}$ is nonhomogeneous Poisson process (in $\mathbb{R}_+$) with intensity function $\lambda(t)$, and if an event occurring at time $t$ is classified as type-$i$ with probability $p_i(t),i=1,\ldots ,n$, independently of what have occured before, then the processes $\{N^i_t,t\geq 0\}$ counting the occurences of type-$i$ events are independent nonhomogeneous Poisson processes with intensity functions $\lambda(t)p_i(t)$. My questions are the followings:

  1. In the definition of a nonhomogeneous Poisson process, the function $\lambda(t)$, as far as I understand, it can be any Riemann integrable function. Can it be any measurable function?

  2. For the above splitting result to hold, what type of assumptions do we need related to the functions $\lambda(t),p_i(t)$? I guess the result is correct for the piece-wise continuous functions. I have seen an exercise in a book, where $p_i(t)$ can be any measurable function. Does this splitting result hold for any measurable functions or what is the largest set of functions for which this splitting result holds?

Thank you very much for your help in advance.

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