Measurable Set can be Written as Countable Union of Measurable Sets 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. 
 A: 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.
A: 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.)
