= 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 distribution
$$
N(t)-N(s)\sim\operatorname{Poisson}\left(\int_s^t\lambda(u)\,du\right),\quad0\leq s<t.
$$
Equivalently, 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 $\mathbb P(N(t+h)-N(t)\geq2)=o(h)$. For merely locally integrable rates use the integrated-rate formulation, which avoids a pointwise regularity assumption.
$$
\boxed{N(t)-N(s)\sim\operatorname{Poisson}\!\left(\int_s^t\lambda(u)\,du\right).}
$$
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