Let $\lambda:[0,\infty)\to\mathbb{R}$ be an intensity function in the sense of \ref{def:inhomogeneous-poisson-process-2026b}, where $\mathbb{R}$ is the set of \reftext{def:real-numbers-c54-2026c}{real numbers}. Then there exist a \reftext{def:probability-space-random-variable-2026a}{probability space} $(\Omega,\mathcal{F},P)$ and an \reftext{def:inhomogeneous-poisson-process-2026b}{inhomogeneous Poisson process} $N=(N_t)_{t\ge0}$ with intensity $\lambda$ on it. In particular, for every real $\theta\ge0$ there is a homogeneous Poisson process with rate $\theta$.