Today we start a big chapter about integration.
First of all, non negative functions.
Read the first paragraph, please. Or the second sentence is very important here.
So we develop the theory mainly.
Four of the.
Real numbers Olivia Measurable Sigma Field and leibig measure.
But all the statements that remain valid for a arbitrary abstract measure space Omega F muell.
But there will be examples in the future.
So now, except that we're talking about real numbers and lidbeck measurable Sigma field.
Suppose we have.
Measurable.
Subset here.
It must be really big, measurable, of course.
But then had non negative function Phi.
Which takes only finitely many values. A1A 2AN is a simple function if all the complete preimages.
Of particular values are measurable.
So any simple function is.
Of course, measurable.
And the polybian integral.
Offer a simple function.
'cause of this of this form for that is the definition. I prepared the picture for you.
So particular values A182 and so on.
And the corresponding preimage is capital AI.
So as expected there.
Integral of a simple function is simply.
So the area under the function, but remember or we are talking about?
Simple functions so.
Play big integral of a simple function is just the area under of the graph.
And in this simple case it is.
Some of.
The areas of rectangles.
What next?
If.
So the measure?
Of particular.
Said AI is Infinity.
So then.
For the value of integro.
Maybe plus Infinity.
Incase AI is positive.
And if AI is 0 or then we accept this convention.
For the zero value of the function multiplied by the infinite measure infinitely big measure of the corresponding.
Kapitol 8 is 0. This is just convention.
In the previous formula.
In this formula.
AI cannot be Infinity, but it can be zero and measure maybe Infinity.
Next
example about the direct clear function of this function is called Derek Lee function.
You already met it.
In
Koma Brook.
If the value is 1 only for a rational X.
But then.
Well, the Riemann integral does not exist. This function is not.
Riemann integrable but on the other hand it is simple function, takes only two values.
And.
The total malibag integral equals one triumphs of the Lib measure of operational's.
Which is zero, as you know twice zero times the measure of the compliment.
The second term is 0 by definition, so the total value or the integral of the directly a function is 0.
So Limburg integration is more general, more powerful than Riemann constructions.
Next definition.
If we have non negative measurable function.
Then the integral.
But which is written down in this way, or in that way?
The value of the integral, by definition, is supremum of Y.
And, uh, why?
Of the set E and function F.
Is collection of all values or the integrals of simple functions which approximate function F from below. So here.
Fi is simple.
And F is an arbitrary measurable non negative function.
So on this set, why?
'cause all of his the form.
0 zero or zero Infinity and so on, so supremum.
All of this exists.
Mcan equal Infinity.
So Please remember and accept the definitions. I usually do not write the arguments.
So FDM is the value of the Libyan integral.
So if the domain is interval then sometimes we write lidbeck integro in this way from A to B.
Sometimes I show the arguments ever fix.
So on the picture.
I think you understand that function.
F is something like this.
And.
Fly.
Is the approximation of the function F from below function F can be arbitrary measurable Phi must be simple function.
Basic properties of integrals, all of them follow from the properties of.
For simple functions and from properties so the supremum all of them are more or less obvious if one function is smaller than another one or then integration over the same domain capital A.
For their first function is smaller.
If B is subset of a, then the value of the integral is again smaller.
You can sketch the graphs to understand better or this properties.
If you multiply function by a constant.
Non negative number A or then the integral is multiplied by the same number.
If it is a null set, then the integral is 0.
And if we have two disjoint sets A&B, then integral over the Union.
Equals sum of two integrals over A&B.
A theorem.
Very simple, more or less obvious, but rather important.
If F is non negative, measurable function or then.
It equals zero almost everywhere if and on.
Only if the integral is 0.
So if if it's zero almost everywhere, then everyone function 5.
Equals zero almost everywhere and.
For every function Phi.
Approximating F from below or the integral is 0.
And the supremum of all such integrals over different simple functions is again 0.
But the second step is a bit longer.
And you must understand for this constructions suppose integral is zero and introduce the set of their function is positive. Remember we are talking about non negative functions only.
If X belongs to here, then it is bigger than this or that.
Number of the form of 1 / N.
And if function F is bigger than the 1 / N and then of course X belongs to here. So this said he can be represented as the countable union of subsets.
EN.
Yeah, in is the complete pre image of the interval 1 / N up to Infinity, so collection of all points where F of X is bigger equal 1 / N.
For every fixed end.
Prove, introduce the simple function.
Richard Costa well over N.
If X belongs to EN and 0 otherwise, so clearly this function, Phi N.
Approximates
from.
Billo
so that function.
Other function F.
And integral of this simple function.
Is simple, it takes only two values, so 1 / N.
And the measure of AN.
And plus zero.
In the compliment.
And this is of course smaller than FDM becausw.
Integral FGM is the supremum.
Over all simple functions and we know that the integral is 0.
So as a result we have for this product.
Which cannot be negative.
And it is smaller or equal zero 1 / N is positive, but we conclude that measure malibag measure of EN is 0.
And the measure of this said he is not bigger than the sum of measures of YN.
Uh.
Becausw
becausw he is the union of a end. Of course this sets.
My intersect.
And measure of he is not bigger than the sum of measures which is 0.
We've proved on that.
The measure of this set is 0.
It means that the function F which is non negative equals zero almost everywhere.
No.
We can extend the basic property number one.
If.
Function F is not bigger than G almost everywhere or then, or the integrals again satisfy of this inequality.
Oh, the proof is simple.
You can simply.
Introduce the set.
Where functions.
Satisfy inequality.
So let.
Here.
He said.
Of X.
Search for that.
F smaller than G.
So measure of the complement is 0.
Function F is smaller than G almost everywhere, so they complemented zero, and after you integrate.
Now this functions.
So you can integrate over E.
He is over, everything is good.
So this is smaller.
Then
GDM.
And on the compliment.
You remember or we need to add together?
Integral a.
Intersection compliment.
Well, this.
Said he compliment is now.
So all of this stuff is.
0.
Becausw measure of complement is 0.
And more general observation.
If a statement about the integrals holds for functions, FG and so on that it holds for functions.
Which coincides with F&G almost everywhere.
So sometimes people say that F1 is modification or version or function air.
So if.
Two functions coincide almost everywhere, or the one of them is.
Slight modification of another one.
So this stuff was about.
Definitions and more or less obvious simple properties next time.
We will talk about.
More challenging things, starting with the Fatou lemma.
Thank you for your questions.