I know that the union of countably many sets of measure 0 has measure 0. How about the case of uncountably many of them?
2026-05-04 14:55:36.1777906536
What is the outer measure of the union of uncountably many sets of measure 0.
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You can create a set of any Lebesgue measure by taking uncountable unions. For example, suppose that we wanted a set of measure $1$. Consider the following set: $$ \bigcup_{0\leq x \leq 1} \{x\} $$ Each set $\{x\}$ has measure $0$, but the uncountable union of these sets such that $0\leq x\leq 1$ is the closed interval $[0,1]$, which has measure $1$.
So uncountable unions of measure $0$ sets could be anything!