Let $P$ be the partial ordering whose conditions are subsets of $(0,1)$ of positive Lebesgue measure, ordered by $\subseteq$.
Show that P is $\sigma$-linked.
I need to prove that there exist a countable partition of $P$, such that any two set in the same piece of the partition are compatible, hence there exist $Z\subset (0,1)$ such that $Z\subset A$ and $Z\subseteq B$.
Attempt
Using the Lebesgue Density Theorem, I know that any $A\in P$ will verify that $A\triangle \phi(A)$ is a null set. Where $\phi(A)$ is the set of points where $A$ has density 1, and $\triangle$ is the symmetric difference : $A\triangle B=(A-B)\cup(B-A)$.
I was thinking that I should be able to define an equivalence relation $A\sim B$ iif $A\triangle B$ is a nullset. Hence definition an equivalence class $[A]_\sim$. And inside each class I would have the desire property. But How do I prove that this partition is countable?
Best Answer
For any set $X$ of positive measure there is a basic open set $O$ so that $\frac{\mu(X \cap O)}{\mu(O)} >\frac{1}{2}$ (by the Lebesgue density theorem).
Consider for every basic open set $O$ the set of $X$ satisfying the above. What can you conclude about those?