Proving the Continuity From Below Theorem

[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.

 

[BvU]

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]

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.

 

[BvU]

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 ?

 

[Sai Alonzo]

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?

 

[fresh_42]

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.