Can Measurable Sets Be Written as Disjoint Union of Countable Collection?

[Bashyboy]

Homework Statement

I am working through a theorem on necessary and sufficient conditions for a set to be measurable and came across the following claim used in the proof: Let $E$ be measurable and $m^*(E) = \infty$. Then $E$ can be written as a disjoint union of a countable collection of measurable sets, each of which have a finite outer measure.

Homework Equations

The Attempt at a Solution

I am not really sure where to begin. I have searched through my book and haven't found any theorem/lemma even remotely like this. Is this lemma used part of more general theorem? If so, what does that theorem look like? I did a google search and couldn't find anything.

 

[S.G. Janssens]

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

Is this even true in full generality? I don't think so: What happens when your measure space consists of a single point $x$ such that $E = \{x\}$ has infinite measure?

 

[Bashyboy]

Is this even true in full generality? I don't think so: What happens when your measure space consists of a single point $x$ such that $E = \{x\}$ has infinite measure?

I am not sure...I have just begun studying measure theory. I sure hope it's true, otherwise I am going to need to contact Royden and Fitzpatrick! I checked my book again, and that is the very lemma they are using in their proof.