Questions: 1. What if for all n, it is always a fixed event A_K occurring? as it is true for all n, K could be only infinity. Is this scenario not making sense? but if it is possible then it's not infinitely many of A_k occur? 2. Do we need to impose an order on A_k here? from MT/2 Continuity of measure, the concept of limit on events exists only for increasing or decreasing A_k's. I am trying to understand the connection of limsup of events to limsup of functions. Here, the union piece looks like finding a sup (if increasing) from MT2 and then the intersection piece looks like taking a limit (if decreasing).
1. I think you are over-complicating this. We have events A_1, A_2, A_3, ... each of which may or may not occur. What is the question? 2. Indeed there is the natural order in the indices: A_1, A_2, A_3, ... There is no order required on the sets themselves; they do not need to be increasing or decreasing. You are right that the union (on events) has a flavour of sup (on functions): the union occurs iff at least one of the events occur. Now, this union from n...infty will be decreasing as a function of n (the more n, the less sets in the union, so it can only decrease), this might be the ordering you are missing. Hence the intersection in front is actually a set-limit even if the A's themselves are not ordered.
I'm not using that in this unit, so I have no plans to make a video on it. The Wiki page en.wikipedia.org/wiki/Set-theoretic_limit seems to do a good job explaining these.
this video was so helpful! better than any of 10 previous videos i watched!
I have a question
Ingeneral
Can I say that the union of A sub n is the supremum of all An ?
And the intersection of A sub n is the infimum of An ?
It sounds logical to me but I haven't actually come across supremum or infimum of events before.
Questions:
1. What if for all n, it is always a fixed event A_K occurring? as it is true for all n, K could be only infinity. Is this scenario not making sense? but if it is possible then it's not infinitely many of A_k occur?
2. Do we need to impose an order on A_k here? from MT/2 Continuity of measure, the concept of limit on events exists only for increasing or decreasing A_k's. I am trying to understand the connection of limsup of events to limsup of functions. Here, the union piece looks like finding a sup (if increasing) from MT2 and then the intersection piece looks like taking a limit (if decreasing).
1. I think you are over-complicating this. We have events A_1, A_2, A_3, ... each of which may or may not occur. What is the question?
2. Indeed there is the natural order in the indices: A_1, A_2, A_3, ... There is no order required on the sets themselves; they do not need to be increasing or decreasing. You are right that the union (on events) has a flavour of sup (on functions): the union occurs iff at least one of the events occur. Now, this union from n...infty will be decreasing as a function of n (the more n, the less sets in the union, so it can only decrease), this might be the ordering you are missing. Hence the intersection in front is actually a set-limit even if the A's themselves are not ordered.
Why its not visible
It is for me.
can you please make a video for liminf of events?
I'm not using that in this unit, so I have no plans to make a video on it. The Wiki page en.wikipedia.org/wiki/Set-theoretic_limit seems to do a good job explaining these.
This video has really helped, thanks so much.
That is the most simplest explaination thank you .
Our handwritten script looks almost the same 😁
I feel really sorry for you then. 😄