Can an uncountable family of positive-measure sets be such that no point belongs to uncountably many of them?
Solution 1:
Here is a quick counterexample under the assumption of CH:
Let $\Bbb R=\{r_\alpha\mid\alpha<\omega_1\}$, and let $A_\alpha=\{r_\beta\mid\alpha<\beta<\omega_1\}$. All those are cocountable therefore certainly have a full [Lebesgue] measure.
But if $x\in\Bbb R$ then $x=r_\alpha$ for some $\alpha$ and so $x\notin A_\beta$ for any $\beta>\alpha$, so it only appears in a countable number of the sets.