density (idea)

Half the integers are even. We all know that, but what does that mean? We say that "half the numbers in (0,1) are less than 1/2", because m((0,1/2))=1/2=1/2*m((0,1)), where m(I) is the length or, more generally, the Lebesgue measure of I. But there is no uniform distribution on the integers! So we cannot hope to find some measure μ for which μ(even numbers)=1/2*μ(integers).

Density is an alternative attempt to formalise the statement at the top. For any (large) n, we can say how many integers 1,...,n are even: either n/2 or (n-1)/2. So either 1/2 or 1/2-1/(2n) out of 1,...,n are even. And the limit of this expression for large n is 1/2. We say that the density of the even numbers in the natural numbers is 1/2.

Definition

Let A={a₁,...} be a countable set. Typically A=N={1,2,...}. But note that the particular enumeration chosen influences the concept of density very strongly! Let X⊆A be some subset. The density of X in A is

lim_n→∞ |X∩{a₁,...,a_n}| / n

Replacing "lim" with "lim sup" or "lim inf", we get "upper density" and "lower density", respectively; note that these always exist, and are between 0 and 1.

Easy facts

• X has density iff its lower and upper densities are equal.
• The density of X is ≥ 0 and ≤ 1.
• If X is finite, its density is 0. If we add or remove finitely many elements of X, its density is unchanged.
• Density is finitely additive: if X and Y are disjoint sets which both have density, then X∪Y has density, which is the sum of the densities of X and Y.
• Unlike measure, density is not sigma additive.
• By re-ordering A, we can get any density in [0,1] for any infinite subset X.

Examples

1. The set of even integers has density 1/2.
2. The sets of squares, primes, perfect numbers (and others) have density 0.
3. Not every set has density. For instance,

   X = {1,3,4,9,10,11,12,13,14,15,16,...} = {1} ∪ {2¹+1,...,2²} ∪ ... ∪ {2^2k-1+1,...,2^2k} ∪ ...

   has no density (its lower density is 1/2, as seen by taking n=2^2k-1, while its upper density is 3/4, as seen by taking n=2^2k).

One can view density as a special case of Césaro means, or even summability, of the characteristic function of X.