Chapter 175

The Riemann integral is created to calculate the area of a function in a given interval through integrable operations.

However, the Riemann integral has many drawbacks.

For example, the limit and integral commutation order needs to satisfy consistent continuity, and when there are too many discontinuities in a function, the function will not be Riemann integrable.

In addition, there are problems such as incomplete Riemann integrable function space.

With the development of society, mathematicians urgently need to establish a new integral theory, which can not only maintain the properties of Riemann integrals, but also solve the problems existing in Riemann integrals. It is precisely because of the ratio that the Lebesgue integral theory was born.

As for the difference between Riemann and Lebesgue integrals, Gu Lu gave a very interesting example.

If there is a sum of money, there are now two ways to count the amount of it.

The first method is to count one by one, and then add it up.

The second method is to count the number of banknotes of each denomination, multiply the face value of the banknotes by the number of banknotes, and then add them together to find the total.

Among them, the first method is the Riemann integral, and the second method is the Lebesgue integral.

“…… However, we now have a problem with how to define the number of banknotes of the same denomination. That is, how to define the size of a set!"

Gu Lu stared at the students below who were listening carefully, and spoke in a serious tone.

"This is where the measurement theory you learned comes in handy. Gu Lu continued, "Now that we have a measure and a measure space, then we need to define the measurable function! First of all, we must give a standard for judging the measurable function......

Gu Lu is talking, and the students are listening.

Gu Lu spoke very seriously, and the students listened very energetically.

Even many students listened to Gu Lu's lecture and were fascinated in their hearts.

Because of Gu Lu's lecture style, it completely gave them a different experience.

From shallow to deep, layer by layer.

When the teacher Gu in front of him talked about the module of Lebesgue integral, he did not directly tell them what Lebesgue integral is, what properties it has, how to apply it, and so on.

But actually.

Gu Lu starts with the Riemann integral, which everyone is familiar with, first describes the shortcomings of the Riemann integral, then the measures taken by mathematicians to make up for it, and finally, introduces the concept of the Lebesgue integral!

Next, there is the derivation of the Lebesgue integral.

Step by step, Gu Lu listed the formula on the blackboard while telling in detail.

And Gu Lu compared Lebesgue points and Riemann points to the problem of counting money, which made it easier for students to understand.

The crowd realized they were wrong.

And it's a big mistake.

This substitute teacher Gu is not the kind they imagined before, but the kind of teacher that Teacher Shi casually found perfunctory.

But...... There are really people who have real talent to learn.

It is quite difficult to explain such a complex course as the real variable function to them so difficult to understand.

But the young teacher Gu in front of him did it!

They can understand the content of Mr. Gu's words in a timely manner, and give correct feedback to Mr. Gu's questions from time to time.

There are even quite a few people who are starting to get obsessed with this feeling.

In the past, when they were in a real function class, they didn't spend it in pain and suffering.

After two hours, my brain hurt when I listened.

But when it comes to understanding, I don't understand much of what the teacher says in class.

Now it's different.

Their thoughts now seem to be as smooth as using rejoicing, and the originally boring and obscure number symbols are like listening to a piece of music composed of countless characters under Mr. Gu's explanation.

Indulge and enjoy.

For them, the real function class is no longer a torture, but a pleasant experience.

After that......

Mom doesn't have to worry anymore that I can't understand the solid function class!

Many students thought so.

didn't realize that Gu Lu was just a substitute teacher.

“…… The Lebesgue integral is established to solve the defects of the Riemann integral, so the Riemann integral has the property of Lebesgue integral, and the Lebesgue integral has the property that Riemann integral does not have. In layman's terms, it's yours and mine, and mine is mine!"

Seeing that the class bell was about to ring, Gu Lu made the final lesson summary.

In two classes, Gu Lu finally explained the Lebesgye points to this group of junior students.

"Do you understand?" Gu Lu looked at the audience and asked.

"Understood!" replied in unison.

Gu Lu nodded, then took out the USB flash drive that the teacher gave him yesterday, turned on the projector, and said to everyone, "Yesterday, the teacher also asked us to assign you ten practice questions for you to do in class." But it seems that there should be no time now. ”

"Well, you take a picture of the topic and go back and do it. The next time I go to class, I'll give you the right answers. ”

The crowd agreed in unison.

Then, the whole classroom entered a relaxed and pleasant atmosphere.

Gu Lu shrugged, waited for the bell to ring, and after saying that class was over, he left with his bag.

…………

"Hey, Mr. Gu, didn't the school leaders listen to my solid function class last Friday?"

Gu Lu shook his head gently, "No, this time the school leaders randomly inspected several classes in the School of Mechanical and Electrical Engineering, and it has nothing to do with our School of Mathematics." ”

"That's good, that's good. Teacher Shi patted his chest with palpitations.

"That's right. Gu Lu suddenly remembered something, took out a USB flash drive from his pocket, and handed it to Teacher Shi, "Teacher Shi, your USB flash drive, return it." ”

Teacher Shi took the USB flash drive, and then patted Gu Lu on the shoulder with a smile, "Teacher Gu, this time I really thank you, you have solved my urgent need! ”

Gu Lu waved his hand, not taking this matter too seriously, "It's just a trivial matter." ”

"But......" Gu Lu continued, "I was a little too engrossed in Friday's real function class, and the students didn't have time to do the ten practice questions you gave me. I gave them homework for class. ”

"It's fine. Teacher Shi waved his hand, not caring at all, "I'm going to give them a lesson soon, I'll just tell them about it when the time comes." ”

After chatting with Gu Lu for a few words, since it was about to be class time, he hurriedly left with his bag.

In the real function classroom.

Teacher Shi stepped on the class bell and walked in.

But when the students in the classroom saw that it was Teacher Shi who walked in, their originally smiling faces instantly became bitter and hateful.

"Teacher, why are you?" asked a classmate in surprise.

"Why can't it be me!" the teacher was confused.

That's right, this is the classroom of the real function class, and the students in front of me are also students in the junior mathematics department.

But listening to this tone, it seems that in the expectations of my classmates, it should not be me who should come here!

"So, what about the substitute teacher who taught us last class?" the student asked in an eager tone.

"You mean Teacher Gu Lu?" Teacher Shi replied with a frown, "I had something to do last week, so I asked Teacher Gu to come over to substitute for the teacher." Now of course, I'm back, and Teacher Gu won't come over to substitute for the teacher!"

"That'...... Then, Mr. Shi, can you have something to do in the future?" said a classmate weakly.

Teacher Shi: "???"

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