Let $E$ be a nonmeasurable subset of $[0,1]$ whose rational translations, $E_1, E_2, ..., E_k,...$, are disjoint. Consider the translations of $E$ by rational numbers $r$, $0<r<1$, why does $\sum m^*(E_k) = +\infty$? Why the series cannot be convergent?
2026-04-30 02:58:30.1777517910
Why does a sum of outer measures of nonmeasurable sets equal infinity?
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Lebesgue outer measure is invariant under translation. So all the terms $m^*(E_k)$ are equal. And since $E$ is not measurable, $m^*(E) > 0$. So in the series, all terms are the same positive number.