Good evening, so you already know the photo lemma.
And the monotone convergence theorem is based on the photo leama and is very important.
So.
This theorem says the following. If we have a sequence of non negative measurable functions which are increasing.
This is the key word.
Monotonically to earth. So for any point X the limit equals F and this sequence is increasing.
But then.
It is illegal to swap limit and integral.
So remember, F is the limited friend.
So in this case it is illegal to swap.
Limit and integration.
But the proof is not difficult.
And as I have said, the proof is based on the fact to Lima.
All functions of fairness smaller than Earth.
So we have this inequality for.
The integrals.
As a result, the supremum is again smaller or then Integral Earth.
And finally, the limit of supremum is smaller or equal.
The integral earth.
Sodium soup is smaller or equal than integral F.
No, according to the photo lemma.
Or this integral is not bigger than a liminf?
I remember the formulation or the photo leama.
Liam in is always not bigger than limsup.
So as a result we have integral F.
Is small or equilibrium in infants smaller or equal him soup?
And limsup is smaller or equal integral F.
As it was shown in advance.
So as a result, William in Limsup coincide and in this case just standard limit exists.
Which equals the common value of limb in limsup or people say lower limit upper limit.
And.
The limit exists and equals integral of the limiting function.
More or less obvious corollary.
Here.
Here you can.
Do note.
You can.
Do note.
Some.
FNSGN
well, then on the left.
So of course functions GN.
Increase to or the function.
G Rich is total sum of functions.
So negative functions.
Oh
something from.
Alright, let's write it more accurately from one to N.
And the sequence of functions GM, which is not negative, it increases.
To the limiting function total.
Some of her friends.
And apply the monotone convergence theorem or they have integral.
Limit.
GN
now that is a what is on the left.
Equals limit.
Integral.
Here, JNDM and since.
In their last expression, ever have only fine it.
Number of terms here.
So this is a limit.
Off some.
Integral a friend.
The M.
F sorry again if I.
Up to end.
And, uh, this limit.
What is limit of?
Increasing sequence.
This limit of course exists and equals a right hand side.
So this is just simple corollary from the monotone convergence theorem.
And next is the theorem saying that.
Any non negative measurable function?
Can be approximated.
From below by.
Simple functions.
Which not negative and such that there is pointwise limit for any value of X fi N of X increases to the value air facts.
So the algebraic proof is presented in the election notes.
Well, this approximating.
Simple function has this form.
I remember or this eye is the indicator function.
Vent.
Now this set is the complete preimage of.
Particular.
Smaller enough interval.
But the illustrating picture is.
Here.
I prepared in advance.
So this is particular or value.
Can be.
Key.
Or what do it do in here? Is K + 2?
Over 2 toyon and so on.
Lisa oil use on the vertical lion.
And with the.
The red lines.
Show the intervals aware.
Well shows the values of the approximating function approach five end of X.
So Oh well, function is between.
K.
And K + 1.
But the value of.
Simple function is exactly K over to and so this is just illustrating this.
Formula the value.
KO2 to N.
Is when function is better in.
K&K plus one, so this is the.
Complete preimage of the value.
K.
Or what to do again?
And.
I hope everyone.
Understands complete preimages switchover.
Discussed.
In depth.
Sirchie
gaplus are on.
So half open intervals because we want.
All the complete preimages.
To be disjoint.
The graph of function file in is presented with red lines.
After that.
If you are.
Understand the picture for the proof is.
Not difficult.
So any function is simple. It takes only finite number of values.
Secondly, one can show that.
Now that the sequence phien increases.
So we fix particular arbitrary value or facts.
And to the values of function by N.
Increase.
And all of them are not bigger than F of X.
So the proof is not difficult on any next step or the approximation becomes more accurate.
And in any case or the function F is bigger than five M + 1, which is bigger than five year.
Again, you can sketch the graphs.
To understand better.
But it isn't him.
For example, next time there will be.
For this same.
Level.
In the middle.
Not the same.
In the middle.
And for the next green.
Simple function.
For real or Bruce Lee, increase.
So it will be.
But this same here, and it will become.
Bigger.
On this interval.
And so on.
Again, it was.
This same here and becomes bigger.
On that interval.
Up to here.
They must hold. Algebraic reasoning is in the lecture notes.
And finally they limit of her friend for every fixed value fax coincides with their fax.
So you must of course remember very well what limit is so.
Starting.
Starting from epsilon or you fix arbitrary epsilon or will take capital N.
Bigger then.
For this ratio.
And after that.
For every lowercase and bigger than capital an.
We have this difference.
Being smaller than epsilon, but that is the definition of the limit for any particular value or fax over health justice sequence.
So for real numbers.
And this.
Brief paragraph shows that the limit becausw aerofex.
So this is Theorem says that.
Any non negative measurable function?
Can be approximated.
Arbitrarily.
I'll be trilliam.
Accurately.
But no negative simple functions increasing to air.
Sequence.
Of non negative simple functions.
Which are increasing and the limit is there.
And of course, as you remember, simple functions.
Make the base of the integral.
And so, having in hand such simple functions, so one can.
Calculate approximately for the value of the integral.
Air for function FDM?
And can compute the integral.
As accurately as you wish.
Alright, that is enough for today. Next time we'll talk about.
Arbitrary.
Functions which may be positive or negative.
Thank you for your persons, good luck.