TITLE: What exactly is the "Carathéodory Extension theorem"?
QUESTION [14 upvotes]: I have read four texts introducing a theorem so-called "Carathéodory's Extension Theorem", and they all differ.
Here is the statement of the Carathéodory Extension Theorem in Wikipedia:

Let $\mathfrak{R}$ be a ring of subsets of $X$
  Let $\mu:\mathfrak{R} \rightarrow [0,\infty]$ be a premeasure.
  Then, there exists a measure on the σ-algebra generated by $\mathfrak{R}$ which is a extension of $\mu$.

I like this statement since it is very simple and clear.
However mainstream textbooks don't introduce the Caratheodory Extension Theorem as this.
For example, Royden's Real Analysis (4th edition) defines premeasure as a set function which is finitely additive, countably monotone (is this a widely used term?), which is different from the definition in wikipedia. Then, he proves a theorem so-called Carathéodory-Hahn Extension Theorem. This theorem does not imply the 'Carathéodory Extension Theorem in Wikipedia' but is deeper than that in Wikipedia, in my opinion. Since he defined premeasure differently, I am quite hesitant to memorize this. I don't want to be out of the mainstream. He even doesn't require the domain of a given set function to have an empty set.
Another example, Folland's Real Analysis states Carathéodory's Theorem as follows:

If $\mu^*$ is an outer measure on $X$, the collection $\mathcal{M}$ of $\mu^*$-measureable sets is a σ-algebra, and the restriction of $\mu^*$ to $\mathcal{M}$ is a complete measure.

Even though this is critical in any proof for any kind of Extension Theorem, this is obviously not 'the Carathéodory Extension Theorem'.
What is the Carathéodory Extension Theorem?

It is off the topic, but I have one more question.
Why Royden tries to include set functions whose domain does not contain an empty set?
If the emptyset is not in a domain, can't we just extend a given set function by defining $\mu(\emptyset)=0$? Does this sometimes break structure of a given set? E.g. ring of sets, algebra of sets or whatever.

REPLY [3 votes]: There's basically two ways of stating the theorem:
Either you choose to state all the critical steps one after the other step by step or you crunch them altogether as one big theorem in wildest generality.
In the situation here it comes to state either in steps that every premeasure on semiring extends to a nonempty semiring by patching the empty set to it then Hahn-Kolmogorov extension and then Caratheodory's construction. Or one states that every premeasure on a semiring extends to a complete measure.
In my personal opinion the former is clearer and focuses on the critical steps while the latter one hides it under biggest generality which is not always preferable. Also when proving things you don't try to merge always all in one since it gets messy but split it into LEMMATA!! And that's what the wiki version did ;)