nLab density of a subset

Idea

The density of a subset $S \subseteq X$ is a measure of how “large” the subset is, or what proportion of $X$ one can consider $S$ to take up.

Definition

Density is usually defined for subsets $S \subseteq \mathbb{N}$ of the natural numbers, or other countable sets, but the concept generalises to sets $X$ equipped with an exhaustive countable filtration $\emptyset \subseteq F_1 \subseteq F_2 \cdots \subseteq X$ (i.e., one such that $\bigcup_{n\in \mathbb{N}} F_n = X$), where each $F_n$ is equipped with a measure $\mu_n$ satisfying $\mu_n(F_n) \lt \infty$ and where each inclusion $F_n \to F_{n+1}$ is measure-preserving. We call such a filtration and system of measures $\mathcal{F} = \{F_n,\mu_n\}$ a measured filtration on $X$.

The key example is of a totally ordered countable set $A = \{a_1,a_2,\ldots\}$ equipped with the filtration $F_n = \{a_1,\ldots a_n\}$ and the counting measure. For instance, $A$ could be the natural numbers $\mathbb{N}$, or any countable subset with the inherited ordering, such as the primes.

Notice that any finite set can also be given the structure of a measured filtration, with a filtration that stabilises after finitely many steps, and with the counting measure.

In the other direction, a $\sigma$-finite measure space $(M,\mu)$ can be equipped (by definition) with a measured filtration by measurable subsets, with the induced measures.

Definition

Given a measured filtration $(X,\mathcal{F})$, a subset $S \subseteq X$ with the property that $S\cap F_n$ is measurable for all $n$, the $\mathcal{F}$-density of $S$ is defined to be the number

$$d_\mathcal{F}(S) = \lim_{n\to \infty} \frac{\mu_n(S\cap F_n)}{\mu_n(F_n)} \in [0,1],$$

if this limit exists.

For a finite set $B=\{b_1,\ldots,b_n\}$ equipped with any measured filtration $\mathcal{F}$, the $\mathcal{F}$-density of a subset $S \subseteq B$ agrees with the simple ratio $|S|/|B|$.

Likewise, for any measure space $(M,\mu)$ of finite total measure and $S$ a measurable subset, we can relate $d_\mathcal{F}(S)$ to $\mu(S)/\mu(M)$ for various choices of measured filtration on $M$ (DR: does this depend on the choice of filtration?)

Applications

Density arguments are of importance in counting prime numbers with certain properties, and have shown great utility in proving partial results about the Birch and Swinnerton-Dyer conjecture on elliptic curves (namely that it is true for a set of elliptic curves over $\mathbb{Q}$ with density roughly 67%). In this latter example, there are a number of filtrations on can place on the set of elliptic curves up to isomorphism, which only makes small changes to final calculated densities.

Any $\sigma$-finite measure $\mu$ of a set $M$ is equivalent, as a measure, to a probability measure. This is constructed by considering a filtration arising from a countable disjoint partition into measurable sets of finite measure.

Upper and lower asymptotic density

One can also consider, instead of the limit in the definition of density, which may or may not exist, the lim sup or the lim inf. These give the upper and lower (asymptotic) densities, respectively.

Related pages

• ∞-finite measure

References

• Wikipedia, Natural density (which deals only with the case of sets of natural numbers)

See also

• Wikipedia, $\sigma$-finite measure

A great number of other definitions of densities of sets of natural numbers (clearly generalisable to other countable ordered sets) are given at

• OEIS? wiki, Density