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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About YiMin

This is just a nerd PhD student of Math@UT Austin.

3 thoughts on “Asymptotic Density

  1. How do you add equations in your blog? Is it a feature of wordpress?

  2. YiMin says:

    do not use your google blog, just use sinaapp or wordpress, that is perfect with latex ,there is a plugin for latex.

  3. YiMin says:

    see yihei.sinaapp.com, or you may register one wordpress for yourself.

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