# Asymptotic Density

When I was doing the Probability homework, I found this asymptotic density is kind of fun. It defines some finite-additively measure, and it is not an algebra.

Definition: $\displaystyle d$ is a map from subsets of $S=\mathbb{N}$ to $[0,1]$, where

$\displaystyle d(A)=\lim_{n\rightarrow\infty}\frac{\#A\cap\{1,2,\dots,n\}}{n}$

Consider the collection $\mathcal{A}=\{A\in S| \liminf d(A)=\limsup d(A)\}$, it is obvious that the collection has the property of finite additivity. However, it is easy to verify that $\mathcal{A}$ is not an algebra.

Refer to this.