1. probability space
1.1 Probability basis
1.2 Probability space
2. Filtration
Filtration directly refers to the non-decreasing KaTeX parse error: Undefined control sequence: \sigmma at position 1: \̲s̲i̲g̲m̲m̲a̲algebra family in the "Random Process Theory" by Qian Minping and Gong Guanglu . As shown.
Generally called σ \sigmaσ -Algebraic flow orσ \sigmaσ -drainage
Swash in Martingale Theory F t F_tFt, That is, the Chinese translation of filtration into "filter" or "filter", it is essentially a non- subtractive σ \sigmaσ -Algebra. The earliest source of this term is not very clear (some say it came from describing the process of stock price changes). Sometimes (Ω,F,P,∪F_t) in martingale theory is called "filtered probability space".
This concept is very abstract and topological. The first impression of this filtration is the filter, because it is accompanied by an order relationship.
This is the information I collected. When I saw this concept, I realized, ah, it seems like my life.
The following explains my understanding as a beginner.
F in the probability space is a measurable set class. A set is an event, then the measurable set class represents the sum of "We can form an event to be treated with the existing method".
For example, I now think that international politics is very ridiculous. For me, these two examples are almost impossible to reverse, so it is more convenient to explain the problem. Then it is equivalent to:
Xueba → low cost performance.
International politics → very nonsense.
I can form a view of these two things. But now I still can't form a view of some things, such as: "Random Process" is difficult for me? It may be difficult, it may not be difficult.
For a given moment, does every city have the right to seek rent? It may be affirmative, or it may be negative.
If politicians have strong empathy, is it good for us? It may be, it may not be. Then it is equivalent to:
- "Random Process" → difficult, easy
- There is no right to rent-seeking cities → existence,
- There is no strong empathy politician → good, bad in the traditional sense
For the above three things that we cannot determine, we treat them as random variables and assign them the probability of taking two values according to subjective or experience or equal division. This kind of assignment is inaccurate. We have to assign one only when we have to make relevant decisions.
But we know that there are untestable sets. Not all sets can be defined with a suitable probability. (Vitali's theorem in the real variable function: the set with a measure of non-zero on the real number each has an unmeasured subset)
So we think in reverse: Since we are not sure about the above three things, we simply kick outside the measurable set class F. We don't care about them, and don't make decisions about them for the time being. After all, decision-making is usually not mandatory, and shelving is also a choice.
Is the world good or bad, justice or evil? I refuse to give a rational answer, because we cannot be sure about this.
What percentage of things are there in the world? Some things, such as driving, can be regarded as one thing, or as five things like "walking to the door, driving the door, opening the accelerator, closing the door, and starting", which can be viewed from a microscopic point of view. Several things for Ganges sand (corresponding to the movement of individual microscopic particles).
In contrast, even if there is only one person in the world with enough information and resources, this person has unlimited freedom and imagination, but this person still cannot truly do "everything in the world". Even if people are not restricted by the authorities or the outside world, what they can do is still limited. The world is so big that people only need to live in 100 of them, and it is enough to know about 1,000 of them.
Therefore, we declare that as this person's behavioral freedom increases day by day, for him, there are more and more things that can be known, and the measurable set in his eyes keeps getting bigger.
When I was young, I had good expectations for international politics. When I grew up, although I still had a positive imagination for many things, some things were excluded. For example, international politics.
As the "range of things that can be viewed" continues to increase, this constitutes a filtration.
This is a philosophically very realistic understanding. As for what is Xn adapted to Fn? That is, at the same moment, the decision I have to make will not exceed the scope of what I can see (the implication is that if it exceeds, I will not make those out-of-scope decisions). This is the same as the restriction of "E ∈ M" in the "Real Variable Function" course, which is a kind of preventive nonsense.
3. σ \ sigmaσ-algebras
σ \ sigma σ -algebras is actually a set system, it guarantees that the set here, no matter how to make the cross and complement, and you can do it casually, the result will still be in this system. This is very important for a good definition of operation.
https://www.zhihu.com/question/36392820
https://zhuanlan.zhihu.com/p/38119668