I am not really a professional, but this question has been asked on Math.SE 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.

It is clear to me that I can build a suitable underlying probability space for a 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.