Is Hahn-Kolmogorov theorem a direct result of Carathéodory's extension theorem? Both theorems assume the same condition and conclusion, except that


*

*Hahn-Kolmogorov  theorem extends a premeasure on a field of subsets to a measure on a sigma algebra generated by the field of subsets

*Carathéodory's extension theorem extends a premeasure on a ring of subsets to a measure on a sigma algebra generated by the ring of subsets.
Since a field of subsets is also a ring of subsets, is Hahn-Kolmogorov  theorem a direct result of Carathéodory's extension theorem?
Thanks!
 A: It will depend, of course, on exactly how the theorems are stated. In his book Introduction to Measure Theory, Tao states the  Carathéodory theorem first (1.7.3) but for outer measures, and then uses this to prove the Hahn-Kolmogorov theorem (1.7.8) for premeasures. However, as the proof to 1.7.8. shows, he has to do slightly more than just apply the  Carathéodory's extension theorem s he has stated it to obtain the second result.  
As everyone knows, though, different authors often use the same name for different results. The Wikipedia articles seem to be particularly confusing about the difference between the two theorems. Perhaps the Wikipedia articles were written by different people who learned from different texts; that is always an issue with comparing different articles on Wikipedia. 
A: Yes. As you said, the various versions of extension theorems all extend a premeasure on some family of subsets $\mathcal{F}$ of $\Omega$ to a measure on $\sigma(\mathcal{F})$, and differ in what conditions $\mathcal{F}$ satisfies.

*

*The most general form of Carathéodory's extension theorem that I'm
aware of requires $\mathcal{F}$ to be a semi-ring, i.e., (1). $\emptyset \in \mathcal{F}$, (2). $A, B \in \mathcal{F} \implies A \cap B \in \mathcal{F}$, and (3). $A, B \in \mathcal{F} \implies \exists n, \exists C_{1,2,...,n} \in \mathcal{F}, A - B = \dot\bigcup_i C_i$.  See, e.g., the textbook by Schilling, "Measures, Integrals and Martingales".


*The Wikipedia version of Carathéodory's extension theorem requires $\mathcal{F}$ to be a ring, i.e., (1). $\emptyset \in \mathcal{F}$, (2). $A, B \in \mathcal{F} \implies A \cup B \in \mathcal{F}$, and (3). $A, B \in \mathcal{F} \implies A - B \in \mathcal{F}$.


*The Hahn-Kolmogorov theorem requires $\mathcal{F}$ to be a field (also called an algebra), i.e., (1). $\emptyset \in \mathcal{F}$, (2). $A, B \in \mathcal{F} \implies A \cup B \in \mathcal{F}$, and (3). $A \in \mathcal{F} \implies A^c \in \mathcal{F}$.  This seems to be the version used in the Ash and Dade textbook.
It's easy to show that a ring is closed under intersection (write $A \cap B = A - (A - B))$, and condition (3) for a semi-ring trivially holds or a ring, so a ring is a semi-ring. Also, it's clear that a field is a ring (that happens to also contain $\Omega$).
Therefore, version 1 ("Carathéodory's extension theorem") implies version 2 implies version 3 ("Hahn-Kolmogorov theorem").
