Show Corollary 1.24. Namely, show that a measure space (X, M , μ) is complete if (X, M , μ) is constructed via Carath ́eodory’s theo- rem (Theorem 1.20). Corollary 1.24. Let (X, M , μ) be a measure space obtained via Theo- rem 1.20. Then (X, M , μ) is complete. Theorem 1.20 (Carath ́eodory’s theorem). Let M be as above. We have (1) M is a σ-algebra. (2) ForE∈M,defineμ(E):=ν(E). ThenμisameasureonM

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