I am not really a professional, but this question has been asked on [Math.SE](https://math.stackexchange.com/q/2787474/169789) already and in spite of a bounty it was not answered. 


That made me decide to give it a try here, and I hope that is acceptable.




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It is clear to me that I can build a suitable underlying probability space for a [homogeneous Poisson point process](https://en.wikipedia.org/wiki/Poisson_point_process#Homogeneous_Poisson_point_process).


It is enough to have a probability space $(\Omega,\mathcal A,P)$ with on it iid random variables $X\_1,X\_2,\dots$ having exponential distribution.


So I could do already with $\mathbb R^{\mathbb N}$ applied with product measure.


Then $N\_t$ can be defined as the cardinality of the set $\{n\mid S\_n\leq t\}$ where $S\_n:=X\_1+\dots+X\_n$.


If $A$ is a measurable subset of $[0,\infty)$ then I can define random variable $\hat A$ as the (random) cardinality of $\{n\mid S\_n\in A\}$ and then $\hat A$ has Poisson-distribution with a (multiple of) $\lambda(A)$ as parameter, where $\lambda$ denotes the Lebesgue measure. In that sense it can be called a Poisson process **on base of the Lebesgue measure**.


Now my question:



> 
> How to build up a probability space allowing me to construct a Poisson process **based on an arbitrary chosen measure** $\nu$ on $[0,\infty)$ with the property that $\nu([0,t])<\infty$ for each $t$?
> 
> 
> 


So this means that for a measurable $A\subseteq[0,\infty)$ the random cardinality of $\{n\mid S\_n\in A\}$ has Poisson-distribution with parameter $\nu(A)$. Further if two such sets $A,B$ are disjoint then $\{n\mid S\_n\in A\}$ and $\{n\mid S\_n\in B\}$ must be independent (as is also the case described above).


Thank you for your attention and (valuable) time in advance.



How to construct a Poisson process not based on Lebesgue measure?
Another construction, which does not use the structure of $\mathbb{R}$ and works for a sigma-finite measure $\nu$ on arbitrary measurable space $\Omega$, is as follows: let $\Omega=\bigcup\_i E\_i$ with $\nu (E\_i)<\infty$ and $E\_i\cap E\_j=\emptyset$ for $i\neq j$. For each $i$ independently, sample a Poisson random variable $N\_i$ with parameter $\nu(E\_i)$, then for each $i$, sample $N\_i$ random points in $E\_i$ distributed according to $\nu|\_{E\_i}$, independently of each other and between $i$. Properties of Poisson random variables guarantee that the number of points in two disjoint sets are independent, and the average is given by $\nu$.


