In the paper below (see attached URL) the author states that Poisson measures "can be used to describe a population of random size living in a zero-dimensional space" (page 17). I have several questions to follow here:
1). Population = number of "people". As far as I understand Poisson measures it could describe the dynamics of population, not the population itself , for example the number of children born within a fixed time interval.
2). Why zero-dimensional space? What does the author mean by that? I can't picture the right model.
http://ssdnm.mimuw.edu.pl/pliki/wyklady/kond_skrypt.pdf
2. Zero-Dimensional space allows the author to consider the population like a steady state system: In my understanding this would allow him to consider the population change as ~0. This is done by assuming that demographic details like mortality(MR) and birth rate (BR) to be balanced. MR-BR=0. A zero-Dimensional space is described as a topological space having a base of sets that are at the same time open and closed (clopen) in it.
1. Poisson distribution describes the probabilities involved with events that occur with a constant probability per unit time. Again, we know that Poison distribution is not a continous function at all, but rather measures probabilities of discrete values. As far as I know, every descrete space is a zero-dimensional space.
I find it easier to visualize it in the plane than in the line, so I'll go for the general case discussed a few lines later.
Think about a Poisson point process, to fix ideas think of the locations of all nests of a specific bird species in a forest. The forest is the "locally compact space" or "Riemannian manifold" in the paper. The nests are the "population in a zero-dimensional space".
Zero-dimensional, intuitively, is that its dimension is 0, i.e. it is formed only by isolated points. Nests do not form lines or spaces of higher dimension, they are at a positive distance from each other.
The Poisson measure of a subset A of the forest is the count of how many nests are in A. That would be the population size (for A).
Since we are speaking of stochastic process consisting of nest configurations, the number of nests in A is not fixed: for another configuration of nests, there may be more or less nests in A. That is what is meant by saying that the size of the population is "random".
Thanks, Pedro! Great explanation.
Tseliso, in what way clopen sets are connected with zero-dimensionallity?
Iaroslav, if X=(Y,t) is a topological space, it qualifies as a zero-dimensional space if and only if it has a basis whose sets are all both closed and open (clopen) at the same time.
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