OK, well we discuss situations when the.
Integration and the limit can be swapped.
So there were already examples showing that this is not always the case.
The first theorem, which justifies such swap of limit and integration awards for the monotone convergence theorem. Now we're talking about dominated convergence theorem.
How do you do the statement, please?
So in case.
Absolute value of all FN is smaller than G.
And function G is integrable, meaning that the integral is finite.
Well then it is a legal to swap limit.
And integration.
But they profess.
Not difficult and like previously it is based on the photo lemma.
Firstly, we consider the case when a fan all functions. The fan are not negative.
So for two salamba gives this inequality.
And the target is to show that limsup.
Of a friend is not bigger than Integral Earth.
The difference dream minus a fan is not negative and we apply the photo lemur to this.
Square bracket.
As a result of it see of the falling.
This limit is smaller or equal than integral G minus stream soup.
So very important comment here. Volume in of minus A equals minus limsup.
But your voice, the definition of what elleman forward to him soup are, and.
Try to understand this.
Equality.
So.
On the left hand side we can consider introduce the difference.
Integral G and minus integral of.
Limit her friend so integral G minus integral F.
Is smaller or equal than the right hand side?
So integral G.
Can be cancelled and we see that limsup.
A friend is not bigger than Integral Earth.
So what do we have entirely?
Or we have.
Limsup?
Have a limp soup.
Integral.
Your friend, the air is smaller equal integral.
FGM.
And the photo lemma.
Mentioned at the very beginning.
Gives us this next inequality.
Liminf.
But soup?
And live in.
Satisfy on the Orbus inequality. Limp soup cannot be strictly smaller than elimin.
So as a result, we have this too.
Limsup and Liminf coincide?
And both equal.
Integral FGM.
That is the end of the proof for the first case.
For the general case, when functions FN and F.
Maybe positive and negative.
We have this inequality for a friend.
Meaning that their friend plus G.
Is not negative and smaller than two geofacts and we again.
Can I play?
Look what was proved.
Limit of this?
Integral is equals.
Integral of the limiting function plus G.
No SBTRKT.
Integral G.
DM.
And the result is.
Like this, but what we wanted to prove, limiting the grill. A friend equals integral air.
A couple of examples.
So here we have.
Infinite domain.
But we have functions of friend.
All this.
Type.
And all of them.
Decuries
as.
And increases.
All these functions are dominated.
By this function G.
And this function G is integrable over infinite interval or that is exercise on calculus. So we can post to the limit limit. Integral fan equals integral of the limit.
And clearly.
Limit.
A friend.
When N goes to Infinity.
In this formula.
But the result is of course.
0.
So this is the example showing that go to apply for the dominated convergence theorem.
Another example shows that if.
Conditions so they dominated convergence theorem are not satisfied.
Well then.
Are there limit of integrals may be different from the integral of the limit?
Now this is.
Simple exercise.
OK, have winter with zero Infinity.
And the functions are look like this.
So this is function F1.
Equals a one on the interval zero over the second function.
Means will be equal or one on the next interval and so on.
So function FN.
Vehicles overall.
On this interval.
So when N increases, the functions move to the right, so integral of everyone function equals one obviously.
And the limiter of a fan is like previously.
0.
Becausw if you fix an arbitrary X.
Well then, for big enough.
Anne.
Well and bigger than X.
Earth end of X.
Becoms 0.
So the limiting function is zero, integral is zero and one is not equal to 0.
So this is another example illustrating that.
One can swap.
Integration and limit only if you.
Justify this.
Operation.
So.
You can use monotone convergence dominated convergence.
But if conditions are not satisfied, this inequality may be false.
The next theorem, called Burp Olivia theorem. In fact, it is just corollary from the dominated convergence theorem.
It concerns serious.
If this series of integrals of absolute values is finite.
But then this series converges that is follows from your background on calculus.
Convergence for almost all X, the sum is integrable.
And like previously, you can swap integration and summation.
So here and sometime below.
Are you for the?
Area of integration.
Is not indicated means that we consider.
Arbitrarily fixed here.
That is, no usual mathematical slang. You can meet it in textbooks.
And the proof is based on the dominated convergence theorem. So introduce this function.
This function is integrable.
And of course, find it almost everywhere, like any integrable function. Integrable means of the value of the integral is finite.
So this series converges and.
Introduce function F which is sum of FK.
And we can put zero for all values of X where the series diverges. That is a null set.
For all particular sums.
We have this inequality, so all particular sums. If you like you can denote them as GN.
But they're dominated.
And we can apply the dominated convergence theorem.
So F is sum of.
If K.
Limiting sum so total sum of the series.
Equals limit of particular integrals of particular sums up to N.
So here again, the way.
Can swap the integration and limit.
Here we have find it.
Some some may shun.
Find it some may shun and integration can be swapped always so we have sum of integrals. Of course, assuming that everyone function is integrable, so we swap summation and integral becausw some nation is finally from one to N.
And eventually.
When you pass to the limit, we simply have the sum of.
The whole series infinite series of the integrals, but this is the end of the proof.
Very important.
Observation.
Oh OK, I will talk about it.
Next time.
Thank you for your patience.