Good day so photos lemma.
Sketch off its proof and example.
So.
If we have a sequence of non negative measurable functions on a measurable subset.
Then this inequality is valid.
So students like to swap integration and limits.
But in general it is not legal.
The lower limit of the integral.
Can be bigger or then the integral of the lower limit.
I'd like to remind the lower limit.
Um?
If we have sequence of numbers.
And then.
Well, this infom.
Over a little and bigger than capital in so this sequence capital zed of capital N is nondecreasing.
And.
So it has a limit, so lower limits are always always exist.
Well, they put off of the fat olama.
In the lecture notes, is presented rather briefly.
That is only a sketch. More rigorous proof is in the textbooks. I advise you to read attentively.
For those who are interested in this.
Theoria
first of all notations.
Who will take infimum of?
All functions K bigger than N.
That is another function GN of X.
And F of X.
Which is the lower limit by definition, is the limit of GM limit of infimal.
I prepared.
I got off for you for your birthday understanding. So function F is here.
Function G in.
Is of course smaller than F.
And when?
An increase is.
Or the function GN in crisis and approaches Earth.
Suppose 5.
Is a simple function.
Approximating function F from below.
So function fly is here.
So it is smaller than F.
Everywhere.
And it is simple function taking only finite number of values.
So our target is to show this.
Inequality.
So without loss of generality, we assume that function F.
Is positive.
Not zero.
Big cause if it is zero or that interval can be.
Ignored.
And we introduce.
Another function.
5 bar which is slightly smaller than five.
So 5 bar is slightly.
Decreased function 5 so 5 minus epsilon.
5 minus epsilon if four is positive and.
Just zero if four is 0.
So epsilon is currently a fixed small number.
No, since 5 approximates air from below and five bar is smaller.
Well, we see that five bar is smaller than F.
Function GN.
Introduced above increases to function F.
As I mentioned 5 minutes ago.
So sooner or later for any eggs.
Function GN or will become bigger than four bar Becausw Phi bar is strictly smaller than Earth.
And JN increases to earth. So the end or will be bigger then bigger or equal than the green function for a bar.
So here is just assumption for simplicity.
More general cases are in the in the textbooks.
We assume that there exists such an.
Search capital N for that for all.
Lower case in for all XGN of X is bigger or equal equal 4.
Bar.
So that is.
What is sketched on the graph?
And of course, one can consider also bigger values of N becausw function GN.
Can only increase.
No JN is bigger than for a bar, so we have this inequality for function JN and fry bar.
Equivalently, or the same inequality can be rewritten in this way.
Remember, all this infimal is exactly function GN.
And.
And the infimal
of FK.
Of course, is smaller than.
All FK.
In film over K bigger than N means that we have this inequality for all values of key which are bigger than N.
And remember.
Or we can see the search capital and that for all.
Will use of lower case.
And bigger than kapitol in everything.
Discussed is valid.
So we have the following.
Inequality integral Fire Bar is smaller or equal integral FK for all key bigger than an bigger than capital.
So we can take in from on the right hand side over all key bigger than capital N.
And we still have this.
Inequality.
And of course, one can take even bigger values of capital.
Becauses everything becomes even better.
As soon as capital N increases.
So we can pass to the limit when capital N approaches Infinity.
So we obtain this key formula.
Integral 5 Bar is smaller than this.
Below a limit limit when capital an approaches Infinity.
Billoo we replace capital N with lower case.
To be consistent with.
Textbooks with the definition of the lower limit.
You can always compare our reasoning with.
Well, this picture.
On the left in this expression.
But we have.
Something depending on epsilon. Remember 5 bar or was.
5 minus epsilon.
So if.
Five is bigger than zero. The measure of this set is fine it.
Well then.
But then.
But the integral of five bar.
Approach is integral of file.
Remember when epsilon goes to 0 or for a bar approaches 5.
So in this simple case, we consider the finite.
Measure for the center where fire is positive.
Other cases are again in the textbooks.
So we pass to the limit.
When epsilon goes to 0.
On the right hand side absolutely is not present at all.
So we have this.
Inequality.
Integral Phi is small or equal. Lower limit of integral FN.
That is.
That is exactly what we wanted to prove.
But remember our plan.
We want to prove this inequality.
Here it is.
So all this.
Inequality is proved and the last step is simple.
Integral of function F According to the definition is the supremum overall fi simple functions approximating function earth from.
Below again, look at the picture function. Phi approximates F from the below and the value of integral is the supremum of all such integrals of simple functions free.
And.
And this supremo.
Overall.
Simple functions.
Is not bigger than the right hand side in the established inequality. So remember Phi was an arbitrary.
Simple function.
Which approximates air.
So this inequality is established that is exactly away for two. Lamba.
Integral of the lower limit is not bigger than the lower limit of the integrals.
Airfare was.
Exactly.
Are there a lower limit of a friend?
So the photo Lima is proved with some gaps in the proof, so that is not absolutely rigorous.
This sketch of the proof.
But I would like to give you the idea of this proof and students who do mathematics must understand.
Proves.
So the example.
On the page.
14
It shows that the inequality in the photo lemma can be strict.
Again, it would be useful to sketch graphs.
We have
functions.
We have functions.
A friend of this shape.
On the interval is zero or one.
We have
functions like this.
So they equal an.
When the argument is from.
Zero to 1 / N.
So this is FN.
Off works.
So one can easily see that the integral of function of friend for any value of N = 1 and that is the area.
For the rectangle.
And ilium in this is just normal limit, equals one.
For any value of X.
But they limit or function F N = 0.
Why?
Becausw the value at zero is 0.
And for any eggs, there exists an.
Bigger than the 1 / X.
Search for that.
For value of X.
1 / N is smaller, so for any value of X.
There is such end and their friend or fax will be 0.
So this means.
That the limit.
A friend of X.
Of course it did cause lower limit limin.
And this limit equals 0.
And integral or volume in is 0.
The limit of integrals equals one.
So the inequality in the photo leama is strict in this example.
For two, lemma is just lemon. Auxiliary statement is very useful for the proofs of many other statements about integrals.
Next time we will talk about monotone convergence theorem.
And later on they will be.
There will be.
Bonded.
Convergence theorem and.
All that stuff will be discussed later.
Thank you for your questions.