What exactly is the "Carathéodory Extension theorem"?

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.

The condition for the theorem for Wikipedia and Royden are the same, just phrase in different language. Countably monotone together with finitely additive immediately imply countable additive, so not much different there. Empty set is always in the ring due to intersection of a set with its complement. By finite additivity immediately can conclude that once you extend to a measure empty set is always measure 0.

The conclusions are indeed different. I think the version where extension extend beyond just the $\sigma$-algebra and produce a complete measure is the one more widely use, because all of my text have it. But uniqueness of extension to $\sigma$-algebra seems standard too.

Folland appears to start with outer measure instead of premeasure. Presumably, it is so that the theorem can by applied to any outer measure, not just one that come from premeasure. However, the conclusion is weaker, and will fail to give certain result such as uniqueness, or even the assurance that everything in the ring will be measurable.

The problem is that the most important part of Carateodory theorem (and sometimes of many theorems) is not the exact statement but the main idea(s) in the proof. In this theorem's proof there is a key idea to remember, which is what sets to call measurable for the (complete) measure it defines. These are the sets that are partitioned additively (for the outer measure) by all other sets.

Another idea to remember could be to remember how to build an outer measure. But this is kind of natural, and therefore easy to remember. Learn the proof and you will know all the statements you have seen.