density

Objects will float in liquids with a higher density. The volume of liquid they displace weighs more than the equal, displacing volume. Eureka!

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.

Den"si*ty (?), n. [L. densitas; cf. F. densit'e.]

1.

The quality of being dense, close, or thick; compactness; -- opposed to rarity.

2. Physics

The ratio of mass, or quantity of matter, to bulk or volume, esp. as compared with the mass and volume of a portion of some substance used as a standard.

⇒ For gases the standard substance is hydrogen, at a temperature of 0° Centigrade and a pressure of 760 millimeters. For liquids and solids the standard is water at a temperature of 4° Centigrade. The density of solids and liquids is usually called specific gravity, and the same is true of gases when referred to air as a standard.

3. Photog.

Depth of shade.

Abney.

© Webster 1913.