Good evening.
So we continue with a very big and important chapter about integration.
On the previous lecture we discussed, we proved the bear pit levy theorem.
Hunt here is.
Underly and very important statement, all the theorem.
Remains the same if we replace the big measure.
With any other measure.
On an abstract universal set Omega.
Over there fixed Sigma field.
The integrals are denoted in this usual way.
Oh, meant something very often. I omit the argument so Integral FD mu.
In the future I will mainly use this notation for the standard Riemann integral, which you studied.
In the course of calculus.
And even for this same function F of the Libya integral is traditionally denoted in this way.
The last comment if we have.
A finite measure which is sum of two measures and the function F is mu integrable and then.
Obviously, Integral F Jimmy equals sum of two integrals.
I'd like to show you immediately one sentence.
From the.
Further lectures.
Connection between Riemann and Libya integrals.
If I remen.
Function if a function is Riemann integrable on an interval.
Well then it is automatically Libya integrable and the integrals are the same.
So Libya integration theory is more powerful, so you remember where there exist functions which are not Riemann integrable, but they are Libya integrable.
And this statement number one.
A function is Riemann integrable if and only if it is almost everywhere continuous with respect to alibag measure.
So keep in mind this theorem.
It did. It comes without proof.
If you are interested, look at textbooks.
OK, now example about Bear Polivy theorem.
Calculate this.
Infinite integral.
So note that this function.
Cannot be integrated, so there is no expression.
4.
The integral.
So integral.
No expression.
4.
Integral is 0 T.
XEX
mine is so well with the X.
So on this.
Infinite integral nevertheless can be calculated using the bare point Levy theorem.
So note that.
Now this function can be represented as this series.
Well, this serious converges.
And we will show that.
Any one term?
Can be integrated by parts and the value is 1 over and squared.
So the detailed calculations are presented.
No, according to the Bear Polivy theorem.
Well, first of all we know that do this serious converges.
It means that we can apply the Purple EV theorem.
Which says that this integral is.
Total sum of particular.
Into gross.
When N varies from one to Infinity.
And this series converges, the value is π ^2 / 6.
It is not easy to calculate this.
Serious.
But those who are interested can find all the proofs and Wikipedia. This is called Basel problem.
No integration over real numbers, but with respect to arbitrary measures.
Just an example. Typical example, if you meet something similar, you can refer to.
These calculations.
So suppose we have.
Interval zero π.
And as usual, they boil Sigma field about the measure.
Is defined by the following expressions.
So we have.
05
and do we have function?
Function sign.
And for any of an interval.
From
zero to π / 2.
Or we just introduce?
Measure of the interval is the difference sciendi Minasyan see chances Notley big measure.
The cause, the length of interval is not standard, so the length of.
Interval shiji
is given by this difference.
And similarly for the interval.
Bigger than π / 2. We have science Y minus Andy, so all this.
Generalized lengths of intervals are of course not negative.
And as soon as we have length of interval.
And by the way, the length.
Or mule?
Of a.
She.
Equals.
Mute of a.
Be.
You can take closed or open plus mu of B.
She.
4.
Hey.
Being smaller than be smaller than Siri, so of course such property must be always satisfied in order you want to define a measure.
So now.
So the extension of this measure from intervals.
To the border Sigma Field and also to the Lib Measurable Sigma Field can be done using exactly the same constructions As for the stranded lidberg measure.
Consider now sets out to measure measurable subsets and so on.
And the target is calculate the integral XG mu.
So first observation, the measure is.
Nonatomic meaning that.
The value.
For everyone.
Singleton.
Is 0.
So open interval, half opened or whatever.
Are they all such?
Values of the measure coincide.
So calculate integral X Jimmy over divide interval.
Firstly, we know that.
Integral.
05
XDMU
can be calculated.
In two steps, her boy.
Glass integral.
Halfpipe boy
ex genial.
Why do we do it? Becausw the measure is introduced in different shape.
For the first half interval and for the 2nd.
Half interval.
So calculate the first integral.
We divide of the interval zero half point.
In an equal sub intervals.
Like photos
so here is X.
Zero half boy.
And the function is like that.
And we approximate function.
X.
Revealed.
Stepwise constant functions.
You already did similar exercise.
In assignment.
One or two in assignment to Celsius Pi over.
2IN.
2π
or do in and so on.
So the function is continuous from the right.
Although it is not very important.
So knowing the basic properties of integrals.
Or, you know that this integral.
Equals.
For the simple function, the.
Measure of the interval multiplied by the value of the function.
So the value of the function is piye or what to end.
The value of the measure is signed.
Of the right hand side minus sign of the left hand side of particular interval. And we add together.
Along all intervals. So index I varies from zero to N -- 1.
So here is difference of sign.
Which can be represented as integral of cosine.
Remember the fundamental theorem of calculus.
So your background in calculus is very important for this module.
So as a result we have.
Total integral FN cosine DN.
Becausw the eggs.
Is.
Integration with respect to the X standard Riemann integral coincides with the.
Libya Integro becausw function cosine is.
Of course, continuous, very simple and it is Riemann integrable.
This Berryman integral coincide server the Olivia again to grow.
So this sequence of functions.
Converges to earth.
Moreover, it converges.
Uh.
Well, not.
Monotonically probably, but a friend.
Converges to Earth, which is X.
And.
Of course, for this function is bounded.
Becausw
it is smaller than.
Boy
over 2.
So we can use the dominated convergence theorem.
And.
For the integral of the limiting function.
The limiting the integral of the limiting function XD mil equals.
Limit of particular?
Integrals of the simple functions so it is limit of integral FN cosine.
And here again, one can swap limit and integral becausw.
Oh, this sequence of functions convergence to X, as it was explained.
So here we again can swap the limit and integral and we finish with this expression.
Integral X cosine GM.
Now again, this function is continuous.
And bounded so it is Riemann integrable.
And the value of the integral, as mentioned recently coincides with the.
This variable integral integral coincides with elliptic integral, so this is Libya.
It equals a Riemann integral.
And using your knowledge of calculus, the integral equals half π -- 1.
And of course, one can repeat absolutely the same calculation in the same reasoning for this second half interval.
Have pie to pie.
The value is around plus half pie.
And finally we have to add together these two integrals. The answer to the question is.
The integral equals π.
So we'll discuss this example again in another section on so-called aradon Nicodim theorem.
No, what I want to say.
Is integral with respect to the Dirac measure I remember.
We already discussed this a little.
Here is more accurate.
Presentation.
So consider all the.
Force it to function F and we're in. We're going to show that integral of FG mu equals F of B in case the measure mu is.
Derica measure concentrated at Point B.
Suppose we have.
A simple function which approximates.
Function F from below.
Well then.
OK, the graph can be like this.
But they have.
Function.
Ever fix?
Paris Point B.
And to be approximate this function from below, not necessarily accurately.
But in some way.
So arbitrary simple function which is smaller than air.
So here is F of B.
And it is clear that.
So this interval cannot be bigger than F of B.
So here is F of the.
So eventually
the following formula for the integral of simple function.
All the values of May or the measure.
For intervals.
Witcher aside, so measure is 0. Here measure is 0 here.
But measure is.
Well, this interval equals one.
Becausw of this interval contains point B.
And the total sum, the total value of the integral of this simple function cannot be bigger than ever be.
And remember the value of integral.
By definition, is the supremum of the all integrals of the all simple functions.
Approximating the given function F.
The maximum possible value is effort being.
For example, if this simple function is like follows.
We put.
Simple function to be equal efobi.
For the point X being B and 0 otherwise. So this.
2nd.
Simple function, it looks like this. It takes value.
Therefore, would be and this zero.
Everywhere else.
So this is simple function.
And the value.
All the integral for this simple function is exactly efobi.
And remember that for arbitrary simple function of a cannot obtain anything bigger.
So the supremum.
In this case, maximum.
So premium or false, such integrals of simple functions.
Actually is maximum because it it is attained or we know which function provides the maximal value and the maximal or supremum value equals therefore be.
Convert is enough.
For today, next time will consider a bit more interesting and challenging example.
Goodbye.