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My general question was is it consistent that any uncountable family of less than $\mathrm{non}(\mathcal{N})$ sets, each with positive measure, has an uncountable subfamily $\mathcal{F}$ s.t. $\bigcap ...

Let Bor($X$) = class of all borel subsets of $X$. Cohen algebra is defined as Bor(X) modulo the ideal of meager sets. The Cohen algebra has a combinatorial : it is the unique atomless complete ...

In Theorem 2.7 in the following notes, we seem to assume the following statement. Let $(\Omega,\mathcal F)$ be a Polish space, and $A\in\mathcal F$ an uncountable set. Then there exists a bijection ...