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.

Advertisements

3 thoughts on “Asymptotic Density

DOODLE SOMETH

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google+ photo

You are commenting using your Google+ account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

w

Connecting to %s