I know that the union of countably null sets must be measurable and have measure 0, but what happens when the number of sets is uncountable? For example, the union measure of all single point sets on [0,1] is 1, so the measure changes after taking the union, but is still measurable.
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5Let $S$ be any non-measurable set. Then, $\bigcup_{x \in S} {x}$ is not measurable. – legionwhale Mar 25 '24 at 11:31
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A concrete counterexample is the set of Liouville-numbers. This set is uncountable , but has measure $0$. So , the union of uncountable measure-$0$ sets can still have measure $0$. – Peter Mar 25 '24 at 12:04