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.


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