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I am working through a theorem in Royden and Fitzpatrick's Real Analysis on necessary and sufficient conditions for a set to be measurable, and in their proof they use the following:

Let $E$ be measurable with outer measure $m^*(E) = \infty$. Then $E$ can be written as a disjoint union of a countable collection of measurable sets, each of which have finite outer measure.

How might one prove this. I could use some help on getting started.

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    $\begingroup$ Is this in the chapter about the Lebesgue outer measure, or in the chapter about the general theory of outer measure? Because the claim is not true without some condition of $\sigma$-finiteness on $m$. $\endgroup$ – user228113 Sep 5 '17 at 16:27
  • $\begingroup$ What is your measure space? (I.e., is this Lebesgue measure on $\mathbb{R}$ or not?) Do you believe $E$ can be written as the plain union of a countable collection of measurable sets, each of which have finite outer measure? $\endgroup$ – Eric Towers Sep 5 '17 at 16:28
  • $\begingroup$ @G.Sassatelli This is in chapter 2 called Lebesgue Measure, specifically in theorem 11 in section 2.4 (called Outer and Inner Approximation of Lebesgue Measurable Sets). $\endgroup$ – user193319 Sep 5 '17 at 16:31
  • $\begingroup$ @EricTowers Honestly, I am not sure: I haven't yet gotten comfortable with the material to make any 'intuitive' judgments or arguments. $\endgroup$ – user193319 Sep 5 '17 at 16:32
  • $\begingroup$ @user193319 That theorem deals with Lebesgue measure on the real line. There, we can chop a set up into pieces that are contained in intervals of unit length, each of which must have finite measure. $\endgroup$ – Xander Henderson Sep 5 '17 at 16:33
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I am working out of the 4th edition. Near the top of p. 41, the authors write

Now consider the case that $m^{\ast}(E) = \infty$. Then $E$ may 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.

Note that this passage is from the chapter on the Lebesgue measure on $\mathbb{R}$, and that the particular quote above is part of the proof of a theorem which asserts several equivalent characterizations of the measurability of a set of real numbers $E$. Hence the theorem (and the quoted passage) is about sets of real numbers. This is important, as the real numbers are $\sigma$-finite—the result does not hold in complete generality.

As an example of such a decomposition into disjoint sets of finite measure, take $$ E_k = E \cap [k,k+1), $$ where $k$ ranges over the integers. Since each interval is measurable and $E$ is measurable (Royden and Fitzpatrick are seeking to show that each of four conditions is equivalent to measurability, hence $E$ is measurable by hypothesis), the intersection of each interval with $E$ is also measurable. Finiteness of the outer measure of $E_k$ follows from monotonicity: each set is contained in an interval of unit length.

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Since the setting is Lebesgue measure on $\mathbb{R}$, we may break up the reals into unit intervals.

Let $F_n = E \cap [n,n+1)$ for $ n \in \mathbb{Z}$. Then $\{F_n \mid n \in \mathbb{Z}\}$ is a countable collection of measurable sets ($[n,n+1)$ is measurable, intersections of measurables are measurable) that are disjoint and whose union is $E$.

(I see that Xander Henderson commented to this effect while I was typing.)

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  • $\begingroup$ Jinx! It seems we had the same thought at the same time. ;) $\endgroup$ – Xander Henderson Sep 5 '17 at 16:39

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