Commonly used special cases

2. the locations of realized events are obtained as a random sample, with the inclusion probability in $A_i$ proportional to the intensity function $\lambda(x)$.

The general case, with spatially varying $\lambda(x)$, is called a heterogeneous Poisson process.

With spatially constant $\lambda(x) = \lambda$, we have a homogeneous Poisson process.

$$\mu_{A_i} = E[N(A_i)] = \int_{A_i}{\lambda(x) dx} = \lambda \cdot |A_i|$$

This is also called CSR = complete spatial randomness, which is tied to homogeneous spatial point processes.