Measure, Integration & Real Analysis · Problem 2A.11

Measures — Sheldon Axler

Mathematics — Measure Theory · outer-measure-disjoint-open-intervals

Problem

Prove that if I1,I2,…I_1, I_2, \ldots is a disjoint sequence of open intervals, then ∣⋃k=1∞Ik∣=∑k=1∞ℓ(Ik).\left|\bigcup_{k=1}^\infty I_k\right| = \sum_{k=1}^\infty \ell(I_k).

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