Proving the Continuity From Below Theorem

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Homework Help Overview

The discussion revolves around proving the continuity from below theorem in probability theory, specifically focusing on sequences of events and their properties under countable additivity. Participants are attempting to clarify the requirements for the proof and the application of induction in this context.

Discussion Character

  • Conceptual clarification, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the need to prove that the union of a sequence of mutually exclusive events equals the union of another sequence to apply the Countable Additivity formula. There is uncertainty about how to structure the proof using induction, with some questioning the clarity of the theorem itself.

Discussion Status

There is ongoing exploration of the proof structure, particularly regarding the induction method. Some participants express confusion about the initial steps and the formulation of the induction hypothesis. Suggestions for rephrasing the question for clarity have been made, indicating a need for more precise definitions and context.

Contextual Notes

Participants note the importance of clearly defining the sequences involved and the theorem being referenced. There is an acknowledgment of the complexity of the discussion, with calls for a more organized presentation of the problem.

Sai Alonzo

Homework Statement


Prove the continuity from below theorem.

Homework Equations

The Attempt at a Solution


So I've defined my {Bn} already and proven that it is a sequence of mutually exclusive events in script A. I need to prove that U Bi (i=1 to infinity) is equal to U Ai (i=1 to infinity) to use the Countable Additivity formula.

My prof states that I need to prove the 2nd property of Countable Additivity by Induction but I'm not really sure how to go about.
 
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Sai Alonzo said:

Homework Statement


Prove the continuity from below theorem. I don't see a theorem. Do you ?

Homework Equations


As in the other thread: which relationships apply here ?

The Attempt at a Solution


So I've defined my {Bn} already and proven that it is a sequence of mutually exclusive events in script A. I need to prove that U Bi (i=1 to infinity) is equal to U Ai (i=1 to infinity) to use the Countable Additivity formula.

My prof states that I need to prove the 2nd property of Countable Additivity by Induction but I'm not really sure how to go about.
Induction is something like:
proving a claim is true for n=1​
PLUS
proving that: IF it's true for n THEN it's true for n+1​
 
Sai Alonzo said:

Homework Statement


Prove the continuity from below theorem. If {An} is a monotone nondecreasing sequence of events in A and lim An is in A (n-->infinity) then
the probability of the countable union of An = probability of the lim An, n--> infinity = lim P(An), n--> infinity

Homework Equations


Assumptions
A1 is a subset of A2 which is a subset of A3 and so on.. (monotone nondecreasing sequence)
{An} is in A
lim An is in A, n--> infinity

The Attempt at a Solution


So I've defined my {Bn} already and proven that it is a sequence of mutually exclusive events in script A.
I need to prove that U Bi (i=1 to infinity) is equal to U Ai (i=1 to infinity) to use the Countable Additivity formula.

My prof states that I need to prove the 2nd property of Countable Additivity by Induction but I'm not really sure how to go about.
 
Well, I don't really feel qualified for the contents of the statements, but: do you recognize what I wrote in #2 about the method of induction ?
 
BvU said:
Well, I don't really feel qualified for the contents of the statements, but: do you recognize what I wrote in #2 about the method of induction ?

Yes I understand the concept of induction but more like I don't know how to begin or what to begin the proof with?
 
I suggest you post this question again in a new thread, this time with a complete description in sections 1, 2 and 3. E.g. what Bn are you talking about? And could you cite the theorem? And under point 3, what is your attempt to formulate the induction hypothesis?

This thread so far is a mess and hard to read, i.e. to figure out what it is actually about.
 

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