The proof of Real Analysis by Royden starts: "by the countable subadditivity of outer measure, we may suppose E is bounded", I don't understand why the countable subadditivity of outer measure allows us to ignore the case in which E is not bounded
2026-03-25 06:07:00.1774418820
Any set E of real numbers with positive outer measure contains a subset that fails to be measurable
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We can write $E$ as a countable union of bounded sets (say, $E\cap [-n,n]$ for eacn $n\in\mathbb{N}$). If each of these bounded sets had outer measure $0$, then $E$ would have outer measure $0$, by countable subadditivity. So one of these bounded sets has positive outer measure, and we can replace $E$ with such a set.