 Solution

An Inhomogeneous Poisson process with deterministic, nonnegative locally integrable rate $\lambda (t)$ is a counting process $N$ with $N(0)=0$, independent increments, and increment distributionEquivalently, for continuous rates it is a simple counting process with independent increments and infinitesimal probabilities $\mathbb{P}(N(t+h)-N(t)=1)=\lambda (t)h+o(h)$ and $\mathbb{P}(N(t+h)-N(t)\ge 2)=o(h)$. For merely locally integrable rates use the integrated-rate formulation, which avoids a pointwise regularity assumption.