Thank you very much for this very good series of lectures. I have a couple of questions please, maybe someone can help to clarify them. 1/ Maybe I misunderstood, at 6:30 professor Joydeep Dutta said "every event is actually related to some set in the Borel sigma algebra" I'm confused by this assertion, the fact that X is measurable only guarantees that each element of the Borel sigma algebra is mapped (by the inverse of X) to an element of the sigma algebra U. So, shouldn't it mean that there may exist some elements of X that are not mapped to a subset of R ? 2/ At 27:00, why it's not Y(w) < X(w) ? 3/ At 43:29, why we have to specify ~P(A & B) = P(A & B) / P(B) and not write ~P(A) = P(A & B) / P(B) ? As the B is implicit by the definition of the new probability space. Edit: ~P is defined on a sigma algebra associated to the sample space B, so ~P is defined for subsets of B (A is not necessarily a subset of B).
1. Not if the R-Euclidean space goes to infinity I would think. I think the Borel Sigma-Algebra is just a special way of looking at the real line. The Borel Sigma-Algebra consists of the least amount of all open and closed sets of the reals. Whatever that means, since the reals go to infinity. I think It is defined that way so that it stays consistent with the abstract definition of what a Sigma-Algebra is while simultaneously being measurable in the measure-theoretic sense. Also the inverse image gets mapped back to U which is the universe. So it seems as though you are mapping two different infinities. Therefore, nothing gets excluded. Also, a Sigma-Algebra is nothing but a randomly defined set that contains the least amount of elements that it takes to define the set you are defining. Just my take. I could be wrong. Reason for of Sigma algebra: Sigma algebra is necessary in order for us to be able to consider subsets of the real numbers of actual events. In other words, the sets need to be well defined, under the conditions of countable unions and countable intersections, for it to have probabilities assigned to it. 2. I am not 100% sure but I think X(w)
everything was going great till Lecture 2. This lecture has completely gone over my head :(
have chat gpt open in another tab and pause when you get confused and ask chatgpt questions. It's a grind but you can get through it
@@anamariawil1 professor is too smart, he doesn't sometimes explain basic things, which cause further confusion
This lecture is going over my head
haahah, let me know if you understood it now
This guy is the professor and uncle I never had. Fantastic lecturer!!!!
at 27:36, it shows Y
Yes
It’s so much fun to learn after college from Joydeep sir.. Awesome explanation! Neat Presentation!
Those are terrible explanations. Sometimes, it seems that he does not understand what he is saying.
That book is trash without a professor or another book (like Shreve) btw.
Further confusion, he doesn’t understands it or can’t explain like the first two episodes?
it is really helpful Lecture.
Appreciating your a clear explanation.
❤❤❤
Thank you very much for this very good series of lectures.
I have a couple of questions please, maybe someone can help to clarify them.
1/ Maybe I misunderstood, at 6:30 professor Joydeep Dutta said "every event is actually related to some set in the Borel sigma algebra"
I'm confused by this assertion, the fact that X is measurable only guarantees that each element of the Borel sigma algebra is mapped (by the inverse of X) to an element of the sigma algebra U.
So, shouldn't it mean that there may exist some elements of X that are not mapped to a subset of R ?
2/ At 27:00, why it's not Y(w) < X(w) ?
3/ At 43:29, why we have to specify ~P(A & B) = P(A & B) / P(B) and not write ~P(A) = P(A & B) / P(B) ?
As the B is implicit by the definition of the new probability space.
Edit: ~P is defined on a sigma algebra associated to the sample space B, so ~P is defined for subsets of B (A is not necessarily a subset of B).
1. Not if the R-Euclidean space goes to infinity I would think. I think the Borel Sigma-Algebra is just a special way of looking at the real line. The Borel Sigma-Algebra consists of the least amount of all open and closed sets of the reals. Whatever that means, since the reals go to infinity. I think It is defined that way so that it stays consistent with the abstract definition of what a Sigma-Algebra is while simultaneously being measurable in the measure-theoretic sense.
Also the inverse image gets mapped back to U which is the universe. So it seems as though you are mapping two different infinities. Therefore, nothing gets excluded. Also, a Sigma-Algebra is nothing but a randomly defined set that contains the least amount of elements that it takes to define the set you are defining. Just my take. I could be wrong.
Reason for of Sigma algebra:
Sigma algebra is necessary in order for us to be able to consider subsets of the real numbers of actual events. In other words, the sets need to be well defined, under the conditions of countable unions and countable intersections, for it to have probabilities assigned to it.
2. I am not 100% sure but I think X(w)
The answer to your question may be is that function X maps some elements in sigma-algebra U to sets of measure zero in the Borel sigma-algebra.
Are there exercises and solutions for this course?
Loved this! Amazing prof.
English subtitles please
I had never seen before
What is Sup?
Check math.stackexchange.com/questions/1018350/sup-in-an-equation
Supremum...google up for understanding of Supremum