I want to find a probability space $\Omega$ that represents Poisson process as $$\Pi : \Omega \to \{A \in \mathcal{P}{(\mathbb{R^+})}\mid |A| = \aleph_0\}$$ Which is a mapping from $\Omega$ to all countable subsets of the set of non-negative real numbers.
To put it in words, once a state of world $\omega \in \Omega$ is obtained, all the points of arrival on time line is determined.
How can I construct this probability space $(\Omega, \mathcal{B},\mu)$
The purpose of this construction is that I want to evaluate integrals taking the form as $$\int_{\Omega}\sum_{x \in \Pi(\omega)}f(x)d\mu$$ in which $f(x)$ can be interpreted as a payoff incurred at time $x$. Or more generally, integrals like $$\int_{\Omega}g(\sum_{x \in \Pi(\omega)}f(x))d\mu$$
One can use product probability space to model a Poisson process. Let $$\Omega = \prod_{k = 1}^\infty \mathbb{R}^+, \mathcal{F} = \sigma\left(\prod_{k=1}^\infty\mathcal{B}(\mathbb{R}^+)\right)\text{ and }\mu = \prod_{k=1}^\infty \mathrm{Poisson}(\mathbb{R}^+),$$ where $\mathcal{B}(\mathbb{R}^+)$ is the Borel $\sigma$-algebra on $\mathbb{R}^+$ and $\mathrm{Poisson}(\mathbb{R}^+)$ is the Lebesgue–Stieltjes measure on $\mathbb{R}^+$ such that $\mathrm{Poisson}([0,x])=1-e^{-\lambda x}$ for all $x \ge 0$.
Now define $\Pi\colon \Omega \to \mathcal{P}_{\aleph_0}(\mathbb{R}^+)$ by $$\Pi(\omega) = \Pi(\omega_1, \omega_2, \dots) = \{t : t = \omega_1 + \dots + \omega_n, n\in \mathbb{N}\}.$$