Let $M=(M_u)_{u\ge0}$ be a \reftext{def:inhomogeneous-poisson-process-2026b}{homogeneous Poisson process} with rate $1$ on a \reftext{def:probability-space-random-variable-2026a}{probability space} $(\Omega,\mathcal{F},P)$, so that the mean function of $M$ is $\Lambda_M(u)=u$ for all $u\ge0$, and let $\lambda:[0,\infty)\to\mathbb{R}$ be an intensity function with mean function $\Lambda$, in the sense of the same definition, where $\mathbb{R}$ is the set of \reftext{def:real-numbers-c54-2026c}{real numbers}. Define

Then $N=(N_t)_{t\ge0}$ is an \reftext{def:inhomogeneous-poisson-process-2026b}{inhomogeneous Poisson process} with intensity $\lambda$ on the same probability space.