I am studying the 'Real Analysis' by Royden, 4th edition.

In page $41$, Royden quote the following:

If $m^*(E) = \infty$ and measurable , then $E$ can be expressed as the disjoint union of countable collection $\{ E_k \}$ of measurable sets, each of which has finite outer measure.

How to prove the above statement?


This statement is essentially equivalent to the $ \sigma $-finiteness of the measure space. Simply take it to be the disjoint union

$$ E = \bigcup_{n= -\infty}^{\infty} E \cap [n, n+1) $$

  • $\begingroup$ Why we need $m^*(E) = \infty$? $\endgroup$ – Idonknow Mar 1 '17 at 15:21
  • $\begingroup$ @Idonknow You don't, but if the outer measure of $ E $ is finite; then the disjoint union can be taken to consist of just the single set $ E $. $\endgroup$ – Starfall Mar 1 '17 at 15:22
  • $\begingroup$ do you mean $E =E $? $\endgroup$ – Idonknow Mar 1 '17 at 15:44

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