Let $\omega$ denote the set of non-negative integers and let $[\omega]^\omega$ be the collection of infinite subsets of $\omega$.


If $S\in [\omega]^\omega$ and $A\subseteq \omega$ we say that $A$ is *well-balanced with respect to* $S$ if $$\lim\_{n\to\infty}\frac{|A\cap S\cap \{1,\ldots,n\}|}{|S\cap\{1,\ldots,n\}|+1} = 1/2.$$ Intuitively speaking, this means that $A$ contains "half of the members of $S$" (which also implies that $A$ is infinite.)


**Question.** Given ${\frak S}\subseteq [\omega]^\omega$ with $|{\frak S}| = \aleph\_0$, is there $A\in[\omega]^\omega$ such that $A$ is well-balanced with respect to every member of ${\frak S}$?



A balancing property of infinite subsets of $\mathbb{N}$
By the strong law of large numbers, if $S$ is an infinite subset of $\omega$, a random subset of $\omega$ will be well-balanced with respect to $S$ with probability one.


By the countable additivity of Lebesgue measure, if $\mathfrak S$ is a countable collection of infinite subsets of $\omega$, a random subset of $\omega$ will be well-balanced with respect to every member of $\mathfrak S$ with probability one.


If Lebesgue measure is $\kappa$-additive (the union of fewer than $\kappa$ measure zero sets has measure zero) then the same holds for a collection $\mathfrak S$ of infinite subsets of $\omega$ with $|\mathfrak S|\lt\kappa$.


