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I am studying the proof of a theorem in Real Analysis by Royden & Fitzpatrick. I don't understand one of the statements.

Let $E$ be any set of real numbers. Assume $E$ is measurable.

Why, if $E$ has an infinite Lebesgue outer measure, can it be expressed as the disjoint union of a countable collection $\{E_k\}_{k=1}^{\infty}$ of measurable sets, each of which has finite outer measure?

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Can you express $\mathbb R$ as the union of a countable collection $\{J_k\}_{k=1}^\infty$ of disjoint measurable sets, each of which has finite measure? If you can do that, then the sets $E_k=E\cap J_k$ will do the trick.

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