Consider $ \Omega=\mathbb{N}.$ Is said that a $E\subset\mathbb{N}$ has a density limit if the following limit exists:

$$\rho(E)=\displaystyle \lim_{n\rightarrow \infty} \dfrac{\#\{k\in E: k\leq n\}}{n}$$

Let $\mathbb{F}$ the class of sets that have density. Is $\mathbb{F}$ an algebra?


No, because $\mathbb F$ is not closed under even finite intersections. Hint: Carefully construct a strictly and quickly increasing sequence $\{x_k\}_{k\in\mathbb N}$ of natural numbers and define $$C_k\equiv\{m\in\mathbb N\,|\,x_k<m\leq x_{k+1}\}$$ for each $k\in\mathbb N$. Let $A$ be the set of positive even integers in $C_1\cup C_3\cup\ldots$ together with the set of positive odd integers in $C_2\cup C_4\cup\ldots$. Moreover, let $B$ be the set of positive even integers.

Exercise: Show that $\{x_k\}_{k\in\mathbb N}$ can be constructed in such a way that $A,B\in\mathbb F$ but $A\cap B\notin\mathbb F$.

Source: Billingsley's Probability and Measure, Exercise 2.18.

  • $\begingroup$ Consider an example sequence $x_k = 1000, 3000, 7000, 15000, \cdots, $. This would cause $\rho_n(A \cap B)$ to oscillate so that it never converges (where $\rho_n(E)$ is the $n^{th}$ term of the sequence under the limit). $\endgroup$ – Sudheer Jun 9 '15 at 12:29

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.