Time Change of the Homogeneous Poisson Process

Let $M=(M_u)_{u\ge0}$ be a homogeneous Poisson process with rate $1$ on a 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 real numbers. Define

$$N_t=M_{\Lambda(t)}\qquad(t\ge0).$$

Then $N=(N_t)_{t\ge0}$ is an inhomogeneous Poisson process with intensity $\lambda$ on the same probability space.