After reading the first paper, I think I understand what the density of a set means in this context. I'll try to illustrate the meaning with a simple example (
@Dr.Peterson please correct me if I'm wrong)...
Let the universal set, U, contain all the positive integers {1,2,3,4,5,6...}
Let set B contain all positive EVEN numbers (this is different to the set discussed in the paper, but this is simpler). So B contains {2,4,6,...}
Then the density of B is given by:-
[math] \lim_{m \to \infty}\left( \frac{1}{m}\mu\{n \leq m \mid n \in B\} \right)[/math]
If m=7, then...
[math] \frac{1}{m}\mu\{n \leq m \mid n \in B\}[/math]
=[math] \frac{1}{7}\mu\{n \leq 7 \mid n \in B\}[/math]
=[math] \frac{1}{7}\mu\{2,4,6\}[/math]
The counting function will return the number of elements in the set (I think) so...
=[math] \frac{1}{7} \times 3[/math]
=3/7 ≈ 0.429
In the limit, as m approaches infinity, you can probably see the density approaches 1/2.
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So the density of a set (compared to the universal set) tells us what proportion of the "possible elements" are included in that set. So for the set of integers that can be divided by 5, with no remainder, the density would be 1/5.
You don't always have to consider the lower limit as 0 and the upper limit as infinity. A regional density is also possible. You might be interested in this page,
the prime number theorem, that mentions the density of the primes.
