Existence of the Inhomogeneous Poisson Process

Let $\lambda:[0,\infty)\to\mathbb{R}$ be an intensity function in the sense of Stochastic Process, Independent Increments, and Inhomogeneous Poisson Process, where $\mathbb{R}$ is the set of real numbers. Then there exist a probability space $(\Omega,\mathcal{F},P)$ and an 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 $\theta$.