Is it seldom necessary to concretely compute the measure of a set in the theory of real functions?

You also said it most of the time.

That's true.

When you need to do the math of a concrete number or an expression. For example, when you are in points. That is to say that it is a measure (lebesque or riemann).

A measure of a set is a systematic way to give an appropriate number to a subset of the set.

Or like when you're counting the probability of something. Probability itself is a measure (total weight is 1).

Like you're still 2 coins, sample space is right, what can you say to these subsets? Can you measure the likelihood of them happening? Hehe.

Because measurement theory is a set of theories, theories talk about universality, that is, without specific calculations, a certain conclusion can be drawn through proof that is applicable to certain situations. So, most of us only care about whether something is measurable or not. And then through this, we can continue to prove other conclusions.

The reason why it's important for a 0 measure set is that most of the features don't differ because two things (but anything measurable) are different on a 0 measure set. The simplest is the lebesque integral: for two functions that differ only on a set of 0 measures, then their lebesque integrals are the same (assuming they are measurable).

For example, if you throw a dart at a target (assuming you never miss the target), this means that the probability that you will still fall on the target is 1But for any n points on the target (note that n is finite, so these points are a finite set), you still have a probability of 0 at any of these points(Actually, any finite set is a 0 measure set).

Hope it helps. I'm just going to talk about it, and I'm going to learn it systematically.

The branch of mathematics that uses real numbers as independent variables is called real variable function theory. It is a further development of calculus, and its basis is point set theory. The so-called point set theory is a theory that specializes in the study of the properties of sets formed by points, and it can also be said that the theory of real variable functions is the study of some of the most basic concepts and properties in analytical mathematics on the basis of point set theory.

For example, point set functions, sequences, limits, continuity, differentiability, integration, etc. The theory of real variable function also studies the classification and structure of real variable functions. The theory of real functions includes the continuous properties of real functions, the theory of differentiation, the theory of integration, and the theory of measures.

Using the countable additivity of the measure, the monotonicity, and a reasonable split of a-b, we can conclude that m(a-b)>=ma-mb

Let a and b be the measurable set, and mb=0, then m(a-b)=ma are detailed in the reference materials.

m(a-b)> or =ma+mb should be m(a-b)> or =ma-mb measure is actually a set to [0,r), which must satisfy the usual basic relationship of length. m(a-b)> or =ma-mb

This is the sub-additivity of the measure, and the measure that satisfies the additivity (if it also satisfies other properties) cannot be found, so it can only be relegated to the sub-additiveness.

ma》=m(a-b)》=ma+mb=ma so is equal to.

In addition, ma" = m(a-b) is the monotonicity of the measure.

If the Lebeguel measure of a set is infinite, then the set itself must be infinite. Because the Lebegus measure of a finite set is zero.

In the theory of real functions, measures m(a-b).

Measure theory is one of the core contents of real function theory, which must be learned.

The remainder set (the sum of the intervals removed each time) measures 1, while the [0,1] interval measures 1, so the Cantor set measure is 1-1=0

A line is divided into three sections to dig out the middle, leaving 2 3, and then each section is divided into three small sections, digging out the middle, leaving 4 9 ,......Then after n operations the remaining length is 2 3 n, so the Cantor set measures it to the infinity power of 2 3, so the measure is 0.

In short, the Cantor set is to remove the middle 1 3 from the [0,1] interval, and then remove the middle 1 3 from each remaining interval, and so on and so on. Because each time the 1 3 of all the remaining measures is removed, after each operation, the measurement is exactly reduced to the original 2 3. Because what is left after each operation is a bunch of inter-cell unions, this group of inter-cell unions is obviously a measurable set.

So we get a descending sequence of a measurable set, the limit of which is the Cantor set. Remember that after doing an operation on the whole interval, the set of the intervals with the length of 1 3 on the left and right is the first term of the sequence. Since the first measure of the sequence is 2 3, the measure of the limit of the measurable sequence is equal to the limit of the measure, and the measure of the descending series is a proportional series of 2 3.

So, the measure of the limit is 0