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An additional, loosely related, question out of curiosity - it is known that if we take any family of less than $\mathrm{add}(\mathcal{N})$ (which denotes the additivity number, i.e. the minimal cardinality of null sets such that their sum is not null) sets of full measure, then their intersection also has the full measure. It is obvious that this statement does not hold if "full" is replaced with "positive (outer)" in the above - simply take the sets $[k, k+\epsilon]$ for some fixed small $\epsilon$, and for all integers $k$ - the intersection of this countable family is empty. But what about uncountable collections of strictly less than $ \mathrm{add}(\mathcal{N})$ sets of positive (outer) measure: can there be such families with null intersection?

An additional, loosely related, question out of curiosity - it is known that if we take any family of less than $\mathrm{add}(\mathcal{N})$ (which denotes the additivity number, i.e. the minimal cardinality of null sets such that their sum is not null) sets of full measure, then their intersection also has the full measure. It is obvious that this statement does not hold if "full" is replaced with "positive (outer)" in the above - simply take the sets $[k, k+\epsilon]$ for some fixed small $\epsilon$, and for all integers $k$ - the intersection of this countable family is empty. But what about uncountable collections of strictly less than $ \mathrm{add}(\mathcal{N})$ sets of positive (outer) measure: can there be such families with intersection of every uncountable subfamily being null?

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

(1) Is it consistent that each uncountable family of strictly less than $\mathrm{non}(\mathcal{N})$ sets of positive (outer) measure has an uncountable subfamily (a) of same size, or (b) smaller, yet uncountable, with an intersection of positive measure?

(2) Is it consistent that each uncountable family at least $\mathrm{non}(\mathcal{N})$ sets of positive (outer) measure has a strictly smaller, yet uncountable subfamily with an intersection of positive measure?

(2*) [aadded in light of the comment of Wojowu referring to some classical ancient constructions] The same as (2), but restricting attention to Lebesgue measurable sets only, i.e. forget about "outer" in the question.

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. $\bigcap \mathcal{F}$ has positive measure?

[reformulated in light of the comment of Wojowu referring to Łuzin-Sierpiński construction that actually answers the questions if we do not restrict to measurable sets]

(1) Is it consistent that each uncountable family of strictly less than $\mathrm{non}(\mathcal{N})$ measurable sets of positive measure has an uncountable subfamily (a) of same size, or (b) smaller, yet uncountable, with an intersection of positive measure?

(2) Is it consistent that each uncountable family at least $\mathrm{non}(\mathcal{N})$ measurable sets of positive measure has a strictly smaller, yet uncountable subfamily with an intersection of positive measure?

An additional, loosely related, question out of curiosity - it is known that if we take any family of less than $\mathrm{add}(\mathcal{N})$ (which denotes the additivity number, i.e. the minimal cardinality of null sets such that their sum is not null) sets of full measure, then their intersection also has the full measure. It is obvious that this statement does not hold if "full" is replaced with "positive (outer)" in the above - simply take the sets $[k, k+\epsilon]$ for some fixed small $\epsilon$, and for all integers $k$ - the intersection of this countable family is empty. But what about uncountable collections of strictly less than $ \mathrm{add}(\mathcal{N})$ sets of positive (outer) measure: can there be such families with null intersection?

(2*) [aadded in light of the comment of Wojowu referring to some classical ancient constructions] The same as (2), but restricting attention to Lebesgue measurable sets only, i.e. forget about "outer" in the question.

An additional, loosely related, question out of curiosity - it is known that if we take any family of less than $\mathrm{add}(\mathcal{N})$ (which denotes the additivity number, i.e. the minimal cardinality of null sets such that their sum is not null) sets of full measure, then their intersection also has the full measure. It is obvious that this statement does not hold if "full" is replaced with "positive (outer)" in the above - simply take the sets $[k, k+\epsilon]$ for some fixed small $\epsilon$, and for all integers $k$ - the intersection of this countable family is empty. But what about uncountable collections of strictly less than $ \mathrm{add}(\mathcal{N})$ sets of positive (outer) measure: can there be such families with null intersection?