TL;DR: At 6.34 how do we know that Aj\A belongs in the semiring? It should in order to apply additivity. I have been studying your videos on measure theory. They are very good. I have one question here. At 6.34, you apply additivity for the pre-measure μ and do: μ[(Αj∩A)U(Aj\A)] = μ(Αj∩A) + μ(Aj\A). However, in the additivity property of the pre-measure --> there was the requirement that both the separate sets and their union --> should belong in the semiring --> because a semiring is not closed under union. Let's check that. Aj belongs in the semiring and thus --> (Αj∩A)U(Aj\A)=Aj belongs also. Next Aj∩A belongs in the semiring because Aj and A belong in the semiring and a semiring is closed under intersection. My question is how do we know that Aj\A= Aj∩A' belongs in the semiring? Semiring is not closed under complement. And Aj, A are totally irrelevant with each other, since they were picked arbitrarily. Am I missing something?
Good question! At this point, I was a little bit sloppy to make the whole proof less complicated. Actually, we have to extend to semiring to a ring such that we can do that calculation. The given pre-measure extends uniquely to a pre-measure on the ring.
How can you prove that a pre-measure on a semiring of sets is sigma sub-additive, without using Caratheodory's extension theorem? Let me explain myself: For a pre-measure on a ring (rather than a semiring) of sets (which is closed under finite union and relative complement), the sigma sub-additiveness is easy to prove, at the condition that we restrict ourselves to the case where the (countably infinite) union of the sets in the cover is in the ring of sets. In fact, given a set A in the ring, let (Aj) be a cover of A (not necessarily pairwise-disjoint), such that each Aj is in the ring; by hypothesis, the (countably infinite) union of such Aj's is in the ring, hence m(UAj) exists (where m denotes the pre-measure); we can construct a new pairwise-disjoint cover of A (whose elements we call Bj's) such that UBj=UAj, by appropriately combining the Aj's through finite union and relative complement, and such Bj's will also be in the ring (hence, again, m(Bj) exists); thus, by using the monotonicity and the sigma additiveness of the pre-measure, together with the fact that, by construction, m(Bj)
Thank you so much! A wonderfully clear exposition which rounds off the series perfectly.
I am happy to help :) And thanks for your support!
I love your videos, thanks for existing here on RUclips
Thanks for the support!
The end of mesure theory? My favorite series
Nice. I have more in other series. See description :)
Well, got more things to study about here.
TL;DR: At 6.34 how do we know that Aj\A belongs in the semiring? It should in order to apply additivity.
I have been studying your videos on measure theory. They are very good. I have one question here. At 6.34, you apply additivity for the pre-measure μ and do: μ[(Αj∩A)U(Aj\A)] = μ(Αj∩A) + μ(Aj\A).
However, in the additivity property of the pre-measure --> there was the requirement that both the separate sets and their union --> should belong in the semiring --> because a semiring is not closed under union. Let's check that.
Aj belongs in the semiring and thus --> (Αj∩A)U(Aj\A)=Aj belongs also. Next Aj∩A belongs in the semiring because Aj and A belong in the semiring and a semiring is closed under intersection.
My question is how do we know that Aj\A= Aj∩A' belongs in the semiring? Semiring is not closed under complement. And Aj, A are totally irrelevant with each other, since they were picked arbitrarily. Am I missing something?
Good question! At this point, I was a little bit sloppy to make the whole proof less complicated. Actually, we have to extend to semiring to a ring such that we can do that calculation. The given pre-measure extends uniquely to a pre-measure on the ring.
How can you prove that a pre-measure on a semiring of sets is sigma sub-additive, without using Caratheodory's extension theorem?
Let me explain myself:
For a pre-measure on a ring (rather than a semiring) of sets (which is closed under finite union and relative complement), the sigma sub-additiveness is easy to prove, at the condition that we restrict ourselves to the case where the (countably infinite) union of the sets in the cover is in the ring of sets.
In fact, given a set A in the ring, let (Aj) be a cover of A (not necessarily pairwise-disjoint), such that each Aj is in the ring; by hypothesis, the (countably infinite) union of such Aj's is in the ring, hence m(UAj) exists (where m denotes the pre-measure); we can construct a new pairwise-disjoint cover of A (whose elements we call Bj's) such that UBj=UAj, by appropriately combining the Aj's through finite union and relative complement, and such Bj's will also be in the ring (hence, again, m(Bj) exists); thus, by using the monotonicity and the sigma additiveness of the pre-measure, together with the fact that, by construction, m(Bj)
I have a community forum for longer math questions, including LaTeX formulas.
Very good series, do you know a source of the proof of the uniqueness, part of the proof with semi-rings ??
Elstrodt has a nice book about that.
Thank You, Man! out there for this masterpiece.
Finally, I completed this Playlist.
Thank You again.
You're very welcome! :) And thanks for your support!
Life saver...