Vol 2 Chapter 1218: Measure theory

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"What is the length? What is the area? What is the volume?"
This unexpected problem left Austrian for a while. He did not expect Wang Qi to ask such questions. After thinking for a moment, he opened his mouth and said, "The length is ..."
The words came to my lips again.
What is long?
I'm afraid everyone knows this problem. Even a newborn demon or a child who has just opened a spirit can have many concepts such as the size and length of the babble.
However, the simpler the concept, the more difficult it is to describe. Because in most people's minds, the concept of "length" is sufficiently basic. Ask them to express such a concept with a more basic concept ...
It's really difficult.
Seeing him, Wang Qi was a little funny, but still said seriously: "Since you can't answer this question, then simply be better. Just ask you, what is the length?"
Austrian Liu can't say anything like "don't underestimate the demon".
He thought about the problem very seriously and finally came to the conclusion carefully: "The distance between two points ..."
"Hehe." Wang Qi smiled.
Oliu was somewhat irritated. He stood up and said, "Who wouldn't understand this self-evident truth? Since everyone understands, then why should we study it? Want me to say that you and other human races have entered into obstinacy obstacles, just staring On his one-third of an acre, he couldn't see the vast sky. "
"Self-explanatory?" Wang Qi's tone was full of ridicule. He used his hands and feet to flip down the lecture table, pulled a somersault in the air, and landed in front of Oulu's desk, staring straight at Oulu. He smiled a bit sullenly: "Boy, I have to tell you that nothing is self-explanatory. The so-called" self-explanatory "is just an illusion, an illusion given to you by your brain, your soul, and your wisdom. "
"The name of this illusion is called" I am born to know ", or" I am very clever "and" My heart is near. "
Wang Qi stood up again: "Perhaps the acquired wisdom will give you a feeling of being born with spirit and wisdom. We can't really understand this feeling. For our human race, 'Linghui 'It's really uncommon. So, we can think that Linghui is not so great-at least the wisdom of our human level is not so great. Everything we know is actually the shackles that wisdom sets for ourselves. "
"Only by recognizing your obsession, you can get rid of it. Only by knowing the smallness of the sky can you calculate the vastness of Xinghai."
Oliu frowned: "Did you sit down and talk? That's pretty good."
In fact, there is some joy in his heart. This kind of conflict between ideas and ideas can be said for a long time, and eventually it may become a situation where no one can convince anyone. If he enters such a situation, he has the opportunity to erase the embarrassing scene just now.
Unfortunately, Wang Qi will not give him this opportunity.
The world is objective after all. True is true, false is false.
Wang Qi turned over backwards, as if to fall to the ground. The next second, he stood on the podium. He said: "Just before you mentioned a set of geometry supported by the" four conditions ", must you also know the" fifth condition "?"
Austrian nodded: "Not absolutely."
"You can call the painting method" condition ", you also have some wisdom to update the demon clan. It's a pity ..." Wang Qi shook his head, and it's a pity that the update demon clan didn't go deeper, or it was a pity that some younger Learning is not good. He said: "Since you know this axiom that our human race calls the sky painting method, you should also know the definition of point, line and plane?"
Austrian nodded. They are indeed aware of such self-explanatory questions.
"The line is the sum of countless points, right?"
Austria Liu nodded again.
Wang Qi continued to ask: "Have you ever thought that the point has no length. Or, the length of the point is zero. The line is composed of countless points, so where does the line come from?"
"Um ..." This question surprised all the students.
Including the two dragons, Yueluo Liuli and Baixuan Suzheng, everyone was stunned.
-This ... No one really thought about it!
Wang Qi looked at the expression of the group of students underneath, and finally nodded in satisfaction.
Small sample, I ca n’t cure you?
I tell you, I look simple here, but the rare question is how much.
The question Wang Qi asked just now is actually an important part of the "measure theory". And its solution is called "Leberg Integral" on the earth. In this world, it is a classic of the sect, completed by the Geting School.
Although this is just a small footnote for the glorious life of the master, most people know that if there are ten thousand disciples who have thoroughly understood this, there will be no obstacles to the period of refining the virtual.
How long does it take for a genius person who has not been exposed to mainstream mathematics to thoroughly understand this problem?
Grothendieck spent three years in high school.
"If you want to explain this problem, you have to start with set theory." Wang Qi said, and began to explain the original content of the set. Set, Zi, element, union, intersection, complement set, etc. ,
All of these contents are for ordinary high school students. About three or five lessons, ordinary high school students on the earth can master this content. The students of Wang Qi are at least high-level demon races in the metamorphosis period, and his memory is more powerful. It only took him one hour to ensure that they mastered the basic concepts of those collections.
Of course, it is really just a basic concept.
"Next, we can introduce some more complex concepts, such as 'infinity'. The concept of 'infinity' is also very complicated. I know that this concept is also discussed by some people outside of mathematics. Maybe some of you I have thought about this question-what is infinity. And now, throw away the definition of mysterious and mysterious in your brain. In mathematics, there are only two kinds of infinity. Countable infinity and uncountable infinity. The set of all integers is Called 'countable infinity'. Uncountable infinity is greater than countable infinity, and there is no infinity smaller than countable infinity ... "
"The set of all real numbers is an uncountable infinity. And all the sets that are equipotential to 'the set of all real numbers' are called 'continuum'." Wang Qi smiled a little at this point in his speech: "By the way, There are quite a few things in this system that I proved. In other words, this is what I learned. This must be emphasized. "
Austrian Liu's complexion.
"The story of myself and the" Continuous System Hypothesis ", I will tell you again when I have time. In short, you know this concept. If you want to ask the difference between" countable infinity "and" uncountable infinity ", then Let ’s talk about it briefly-this may be the mystery between a point and a line without a size. "
"Now back to the subject. What do we usually mean by the words length, area and volume?" Wang Qi put his hands on the table: "To clarify the subject more clearly, let us focus on the simplest One-dimensional situation, that is to say, we only consider the word 'length'. We hope that if a part of the line is taken out, there will be a 'length'. If this can be done, then similarly, the area and volume Similar words can be understood similarly. "
"First of all, we can make such a definition. A straight line is a huge set of points."
"Every subset of this point set, including itself, has a 'potential'. This potential is a measure."
"The union of two subsets that do not intersect each other-that is, another subset of this large set of points, also has a measure and this measure should be equal to the sum of the two. In simple terms, two disjoint and non-overlapping The total length of the line segments can be regarded as the sum of their respective lengths. "
"Further, the sum of the measures of the three disjoint subsets should also be equal to the measure of the set of these three subsets combined, either four or five, and so on, the measure of the infinite disjoint subsets The sum should also equal the measure of the set obtained by combining them. "
Speaking of which, Wang Qi solemnly said: "Next, we can make a final definition."
"One, the measure corresponding to the empty set is zero. Two, a number of subsets that do not intersect each other, and the measure of the subset obtained by combining them is exactly equal to the sum of the measures of these subsets. Real number axis, then the measurement corresponding to the line segment from sub point to ugly point on the number axis should be equal to ugly minus sub. "
Then, Wang Qi closed his eyes and looked solemn.
"It's a tight definition of what people commonly call" length ", and it's the only correct definition."
Of course, this sentence is slightly exaggerated. The Lebesgue measure that Wang Qi just described [in this world, it is called the 'Geting measure'] is just one of the measures. In fact, there will be other measurement systems in this world. Mathematics also recognizes other measures that are different from this measure.
For example, another measure that would be involved in geophysics, the Dirac measure [this world is also called "measureless"]
This has led many mathematicians to believe that there are actually more universal measures, but that humans have not yet understood that step.
However, at this stage, the word "length" has been strictly defined, and human knowledge of arithmetic has reached a deeper level.
For a moment, Austrian Liu was really shocked. He marveled at this rigorous wisdom. But he still asked: "What's the use of this? It's just playing word games. I can play this kind of game for a day ..."
"Yeah, what's the use." The figure flashed, and Wang Qi appeared again in front of Austrian Liu: "This definition just makes you aware of your smallness."
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