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.
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!
Well, got more things to study about here.
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!
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.
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.
Life saver...