5
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(Show that)If $\mathscr{U}$ is an ultrafilter on $\omega$, then $\mathscr{U}$ contains a subset $A$ of lower density zero. (But) there is an ultrafilter on $\omega$ such that every $A \in \mathscr{U}$ has positive upper density.

This is an exercise on page 76 of Problems and Theorems in Classical Set Theory, Peter Komjath , Vilmos Totik. I have no clue how to start.

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  • $\begingroup$ While I admit I don't know this subject, I don't see a question here. Do you have to prove that there's an ultrafilter on $ \omega $ such that every $ A \in \mathscr{U} $ has a positive upper density? $\endgroup$ – Waleed Khan Mar 14 '13 at 4:24
  • $\begingroup$ The lack of question marks. These are all statements. $\endgroup$ – Waleed Khan Mar 14 '13 at 4:30
  • $\begingroup$ @Waleed: There are obviously two questions: to show that every ultrafilter on $\omega$ contains a set of lower density $0$, and to show that there is an ultrafilter on $\omega$ whose every member has positive upper density. $\endgroup$ – Brian M. Scott Mar 14 '13 at 5:01
  • $\begingroup$ @BrianM.Scott I didn't think it was so obvious 28 minutes ago. $\endgroup$ – Waleed Khan Mar 14 '13 at 5:02
  • $\begingroup$ @Waleed: So I gathered. I just find that rather surprising. $\endgroup$ – Brian M. Scott Mar 14 '13 at 5:03
5
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The first question (Nr. $6$ in the book), follows immediately from Nr. $5$:

If $\mathscr{U}$ is an ultrafilter on $\omega$ and $0=n_0<n_1<\dots$ are arbitrary natural numbers, then there exists an $A\in\mathscr{U}$ with $A\cap[n_i,n_{i+1})=\varnothing$ for infinitely many $i<\omega$.

For $k>0$ take $n_k=k!$, say. To prove the result in Nr. $5$, let $\langle n_k:k\in\omega\rangle$ be such that $0=n_0<n_1<\dots~$. Let $$A=\bigcup_{k<\omega}[n_{2k},n_{2k+1})\;;$$ clearly $$\omega\setminus A=\bigcup_{k<\omega}[n_{2k+1},n_{2k+2})\;.$$ Both $A$ and $\omega\setminus A$ have empty intersection with infinitely many of the intervals $[n_k,n_{k+1})$, and since $\mathscr{U}$ is an ultrafilter, one of $A$ and $\omega\setminus A$ is in $\mathscr{U}$.

For the second question, let $\mathscr{I}=\{A\subseteq\omega:\overline{d}(A)=0\}$, where $\overline d$ is the upper density. It’s not hard to verify that $\mathscr{I}$ is an ideal, so $\mathscr{F}=\{\omega\setminus A:A\in\mathscr{I}\}$ is a filter on $\omega$. Now just let $\mathscr{U}$ be any ultrafilter containing $\mathscr{F}$.

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