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Both theorems assume the same condition and conclusion, except that

Since a field of subsets is also a ring of subsets, is Hahn-Kolmogorov theorem a direct result of Carathéodory's extension theorem?

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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.

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