Spatial point process and stochastic measure (II.): The Story of Measure

Since this topic topic is about random measures, then nature is inseparable from the concept of "measure" (Measure). So in this article, we're going to say a measure. Perhaps, in the eyes of many friends, "measure" is the concept of a special theory-it seems that only the people who study mathematics should care about it. This may be related to the curriculum design of the university, as it is generally taught in graduate mathematics, such as "real analysis" or "Modern probability theory". And, in most textbooks, its first appearance has been a thick veil-in most of the textbooks I've seen, it's always defined on Sigma algebra, and sigma algebra sounds like a iffy noun.

Measure, actually very simple

Here, I just want to push through the mystery of measurement-in fact, the measure is a very simple thing: understanding it, only need the knowledge of elementary school students, not graduate students.

or return to our example of counting the stars.

In this example, we define a "number of stars" function, denoted by the symbol N. The input to this function is a collection (such as A and b), and the output is a number-the number of "stars" contained in the set. Let's see what the features of this function are. First, it is non-negative, that is, it is impossible to include a "negative" star in an area. Second, it has an "additive" nature. What does that mean?

For example, in the above two disjoint areas A and B, each contains 5 and 44 points. Then the sum of A and B contains a total of 49 points. In other words, N (a U B) = N (a) + N (B).

Strictly speaking, if an "aggregate function", or a mapping from a set to a nonnegative real number, is equal to the sum of the values that it takes on the set of the finite disjoint set, then we consider the function to be "additive". Further, if it conforms to this additive on the set of infinite disjoint sets, then we say that it is "countably" (additive).

A nonnegative "aggregate function", which is called a "measure" (Measure) if an empty set is evaluated as 0, and if there is a column of additive on a series of sets. As an example, the "Number of stars" function above is a measure defined on a subset of all two-dimensional spaces. Similarly, we can cite a number of concrete examples of "measures", such as:

1. Total mass of all Stars in each region
2. Area size of each area

The paradox of non-measured set and split-ball

However, under certain conditions, measures cannot be defined on all subsets. To say the popular point, is to some of these sets, we can not define its measure. For example, in a two-dimensional plane, we can define the area function according to the general understanding, such as the length and width of a and b respectively, the rectangular area of AB. For complex shapes, we can calculate the area by integral. However, is not all of the two-dimensional plane of a subset of the existence of an "area" it? The correct answer seems a bit "contrary to common sense": in admitting the axiom of choice (Axiom of Choice), it is true that there are some sets that do not define the area. Or, no matter how much area we define on these sets, it can lead to conflicting results.

It is important to note that the "area is not defined" and "area is zero" are two different things. For example, the area of a single discrete point or line on a two-dimensional set is zero. And those subsets of "unable to define area"-What we call "non-measurable sets"-are very, very strange sets--for these sets, we define the area as zero, or any other nonzero number, which can lead to a contradiction. Such a collection is a special ingenious way for mathematicians to construct-in real life, we will not encounter. Such a construct is not difficult, but ingenious. Interested friends can find this structure in almost every textbook on measuring theory, which is not detailed here.

(Note: Not I made, but from http://www.daviddarling.info/)
about the non-measurable set, there is a well-known "paradox", called "The Tuskegee-the split-ball paradox" (Banach-tarski Paradox). If some strange collection does not define an area that can be accepted by many people, then "Tuskegee" may make many people "simply unacceptable"-including many famous mathematicians in the 230 's. This "satires" is said:
We can be a three-dimensional radius of 1 solid ball with some ingenious method divided into five equal-five equal to the meaning that one part of the rotation can be coincident with another copy-and then the five pieces of rotation translation, can be combined into two radius of 1 solid ball. Simply put, a ball is split and reorganized into two balls of the same size!
Of course, such a process can continue, the two changed four, four changed eight. It is said that this is obviously not correct, and then he is so argue:
If a solid ball volume is V (because the radius of the ball is 1, so v > 0), then five sub-blocks, each volume is V/5, translation rotation does not change the volume, so, no matter how they are combined, the final thing is the total volume of V, And it can't be 2V.
However, this is true in the traditional sense-you will be praised as a good thinking child when you take it to a high school teacher. But when I look at it in more general terms, there is a problem. Because, this argument is based on the assumption that each chunk has a "volume". The subtlety of the Tuskegee is that it divides the ball into five "non-measurable sets"--five odd blocks that "cannot define volume." So, here we say that the "five halves" are just saying that one of them can overlap on the other when they are rotated, not that they are "equal"-because there is no volume and there is no equality.
This technique can be achieved by the decomposition of free groups within the abstract algebra. For friends who are interested in Tuskegee, you can refer to the

Leonard M. Wapner's Math reader the Pea and the Sun:a mathematical paradox.

The attentive friend may have noticed that the structure of the non-measurable set or the Tuskegee is based on the recognition of the "Axiom of choice" (Axiom of Choice). If we don't admit it, is it all right? Before we refuse to recognize the "axiom of choice", we first need to know what the "Axiom of choice" is. In layman's words, the axiom of choice can be described as:
Any set of non-empty sets (possibly infinite numbers), we can pick one element from each set.
It looks very "innocent"-it's not the typical "right crap"-so it's called "Axiom." But it is such an axiom, but the magic is amazing, can let us put solid ball a change two. This is the charm of mathematics!
Historically, Barnabas and Tuskegee the era of the split-ball paradox, which was the time when mathematicians debated the existence and abolition of the axiom of choice. Mathematicians were divided into two factions, one for the "Axiom of choice" and the other to oppose it. While the two mathematical geniuses of Barnabas and Tuskegee were opposed to accepting the axiom of choice at the time, they painstakingly found the method of the goal of denying the axiom of choice with this unacceptable "absurd phenomenon". In the later development, most mathematicians still recognize the importance of the axiom of choice in the development of modern mathematics (for example, the core theorem in functional analysis--hahn Banach extension theorem depends on the recognition of the axiom of choice), and choose to accept it, of course, the Tuskegee "strange phenomenon" is also accepted. Now, the "Barnabas-Tuskegee Split-ball Paradox" is also known as the "Tuskegee-ball-Splitting theorem"-from paradox to theorem.
Mathematics is such a wonderful world. It is often built on the basis of our common sense of life, but once it is established it is necessary to follow its own rules of development, even if it sometimes violates "common sense"-the common sense that people can intuitively perceive is limited, and the power of mathematics can take us where common sense cannot reach.
Algebraic structure of measure and set

The operation of the

measure and set is closely related. According to the definition of the measure, if A and B are two disjoint sets, if the measure of A and B is determined, then the measure of their set is determined, equal to the sum of their respective measures. If B is a subset of a, then if their measure is measured, then the measure of their difference set A–B is determined, which is equal to the difference of the measure of A and B. Therefore, when we want to define a measure, it is often not necessary to define all the sets, as long as a subset of the set is defined, the other sets of measures are determined.
We talked about the set of disjoint sets and the difference set, so what about the general set? If A and B are two possible sets of intersections. Then their assembly a U B can be divided into three disjoint parts: A–c, B–c, and C three parts, where C is the intersection of a and B. As long as we know the measure of intersection C, according to the measure formula of orthogonal set and difference set, we can know the measure of A–c, B–c, and a U B. But only knowing the measure of A and B, the measure of their intersection is obviously uncertain--two even the same size set, may intersect many, even overlap, may not intersect.
So, to effectively define a measure, we first need to determine its value on a series of collections and on all of their intersections. Thus, the measure of all the set and difference sets of these sets is given. Mathematicians have summed up this observation into an algebraic structure-the semiring--of the collection note that this is not the same as the semiring in the abstract algebra. S is a set of sets, if the intersection of any two sets in S is still within S, and the difference set of any two sets in S can be represented as a set of other finite disjoint sets in S, then S is called a semiring. Then, as long as the set of s in the definition of a good measure, then by the set of the number of intersection of these sets and set difference set operation of those generated by the measure of the set is also determined.
A set of collections that, if they contain an empty set, and are closed for a number of intersection sets, the set is actually a sigma algebra. In a sense, if we determine a measure on a semiring that covers the complete set, then the measure of all the sets in the whole Sigma algebra is determined. This makes a less rigorous analogy with linear spaces. In linear algebra, for a linear function, if its function value on the base is determined, then its function value in the whole linear space is also determined. For Measure, semiring is like "base", while Sigma algebra is like the whole space. The math of the
number of stars continues: random measure

Back to the process of counting the stars. As we have discussed above, counting the stars is actually a measure. However, every night we see the distribution of the stars are changing. In other words, every time a star is counted, a different measure is obtained. It's kind of like throwing a dice. Each time we roll the dice, we get a different number--this point can be seen as a random variable, and the value of the variable is an integer from 1 to 6. Similarly, the distribution of stars is uncertain, and each time a different measure is obtained-this can also be seen as a "random variable", except that the value of the variable here is a measure, not a number. Such a "random variable", which is measured as a value, is called "stochastic measure" (Random Measure). This is the story to continue to tell in the next article.

Source: http://blog.sina.com.cn/s/blog_a0e53bf70101lenv.html

From for notes (Wiz)

Spatial point process and stochastic measure (II.): The Story of Measure