Carathéodory's extension theorem extends a premeasure on a ring of subsets to a measure on a sigma algebra generated by the ring.
One popular proof of the theorem is a contructive one (please correct me if I am wrong):
- first extend the premeasure on the ring to an outer measure,
- then construct a measure space from the outer measure, with the measure being restriction of the outer measure on the sigma algebra of the measure space. The measure space is complete wrt its measure, by the way.
- third prove the sigma algebra of the measure space contains the ring, so it contains the sigma algebra generated by the ring. So the sigma algebra generated by the ring and the measure restricted on it will be the resulting measure space in the theorem.
I was wondering if there is a different proof which is also constructive but not using an outer measure? References are perhaps just enough.
When, i.e. under what conditions on the ring and the premeasure on it, will the measure space constructed by restricting "the complete measure space directly out of the outer measure" to "the sigma algebra generated from the ring" happen to be also complete?
When it is, will it coincide with the complete measure space directly out of the outer measure?