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# Monotone Convergence

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1. Monotone Convergence Theorem:

Does this follow straight from the theorem?

Have I got to prove something to show it follows from the theorem? If so, what?
2. It almost follows from the theorem. But first you have to write for some in such a way that is an increasing sequence tending to . (Then you have something that satisfies the hypotheses of the theorem.) Can you do that?

Then the case for is similar by considering positive and negative parts.
3. (Original post by nuodai)
It almost follows from the theorem. But first you have to write for some in such a way that is an increasing sequence tending to . (Then you have something that satisfies the hypotheses of the theorem.) Can you do that?

Then the case for is similar by considering positive and negative parts.
Let where is the characteristic function.

Then since , is an increasing sequence of functions which converges everywhere to .

So the result now follows from the MCT. Right?

Any function can be written as where are the +ve and -ve parts of .

Then we know that if then .

If I let then is increasing and converging everywhere to and let so is an increasing sequence of functions which converges everywhere to .

So and the results for and follow from the MCT.

What is the significance of the question saying converge?

Does the result for now follow from the MCT?
4. (Original post by TheEd)
Let where is the characteristic function.

Then since , is an increasing sequence of functions which converges everywhere to .

So the result now follows from the MCT. Right?
That's right.

(Original post by TheEd)
Any function can be written as where are the +ve and -ve parts of .

Then we know that if then .

You could essentially have stopped here. Since are nonnegative, you could have just said that the result follows from the first part of the question.

(Original post by TheEd)
What is the significance of the question saying converge?
That's just the statement that is integrable for each . It ensures that and are all finite so that you're not taking infinity away from infinity or anything horrible.
5. (Original post by nuodai)
That's right.
OK. So how do I exactly structure the argument?

Let .

Then since we have that is an increasing sequence of functions which converges everywhere to .

Now we are supposing converges.

So by MCT,

How does the results that and now follow from the MCT?
6. (Original post by TheEd)
OK. So how do I exactly structure the argument?

Let .

Then since we have that is an increasing sequence of functions which converges everywhere to .

Now we are supposing converges.

How does the results that and now follow from the MCT?
Let where . Then since .

Then is an increasing sequence in tending to

So by the monotone convergence theorem

and
7. (Original post by nuodai)
Let where . Then since .

Then is an increasing sequence in tending to

So by the monotone convergence theorem

and
OK. So why do we have to assume that the integral of the absolute value of converges for the 2nd part?

Can I split it into 2 integrals like I did before with there?
8. (Original post by TheEd)
OK. So why do we have to assume that the integral of the absolute value of converges for the 2nd part?

Can I split it into 2 like I did before with |.| there?
, and , so if then and , so that the difference is well-defined. This is something you should have come across in the definition of (-)integrability.

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