"Mathematics in machine learning" from sigma algebra, measure space to random variables

Σ algebra

So that X is a subset of the set of all subsets (subsets) of a sample space, the set X is called the Σ Algebra (Σ-algebra) and is called the σ domain (Σ-field).
It has the following properties:

(1) φ∈x; (Φ is an empty set)
(2) If a∈x, then A's complement set a^c∈x;
(3) If ai∈x (i=1,2,... ) is ∪ai∈x;

Measurable space

Ω is an arbitrary set, and X is a set of a subset of Ω that is stripped of the extremes in Ω, so that the rest is a set that can be processed, so (ω,x) is called the measurable space (a measurable set). x satisfies the three properties of Σ algebra, we can define measures for the elements in X, so the elements of X are called measurable sets (measurable set).

Measure Space

The measurable space that defines the measure is called the measure space.
Order (Ω,X) is a measurable space, in X defines an equation ν called a measure (a measure).
It meets the following conditions:
(i) Non-negative: 0≤ν≤∞
(ii) ν (empty set) =0
(iii) If XI∈X, where Xi is disjoint, then ν (∪xi) =σν (Xi).
Then (ω,x,ν) is called the measure space (measure spaces).

Lebesgue measure (Lebesgue Measure)

Mathematically, the Lebesgue measure is a standard way of giving a subset of Euclidean space a length, area, or volume. It is widely used in real analysis, especially for defining Lebesgue integrals. A set of volumes that can be assigned is called Lebesgue, and the volume or measure of Lebesgue a can be measured as λ (a).

• If a is an interval [a, b], then its Lebesgue measure is the interval length b?a. The length of the open interval (a, b) is the same as the closed interval, since the difference of two is 0 set.
• If the interval is [0,1], the Lebesgue measure L ([0,1]) is a probability measure.

Probabilistic space

If ν (Ω) = 1, then ν is a probability measure, recorded as P. (Ω,X,P) is called probabilistic space.
In this way, we can think of p as a measure of the set, linking the set to the probability.

Probability space of probability theory study is a measure space (ω,x,p), where P is a measure defined in X, called probability measure. Set Ω We are generally called the sample space, the elements in X are called observable sets, but we prefer to call events, and X is called event domain. Any element A in X, which is a subset of Ω, is an event, and its measure P (a) is the probability of event A. It can be seen that this ternary group (Ω,X,P) of things are indispensable.

Discussion on measurable space and measure space

We know that any event is a subset of the sample space, but a subset of the sample space is not necessarily an event. In order to discuss convenience, or to use a better understanding of the phenomenon as a metaphor. Suppose to study the sexual orientation of the person, so that the sample space x={male, female, not male, because no male is not good to determine its sexual orientation, so in the study of this situation discharged, only study male and female. In other words, the sample space is ω={all men and women}, is a finite set, its corresponding event domain to take f={ω subset of the entire}, (Ω, F) is a measurable space. The probability measure p on the corresponding f of the sexual orientation problem you say is unknown and needs to be determined by statistical method.
A more common practice is to define a random variable on (ω,f,p), using statistical methods to determine the distribution of random variables rather than the P itself. For example, any ω∈ω, defined x (ω) = 0, if Omega is a monk, x (ω) = 1; If Omega is a nun, x (ω) = 2; If Omega is the husband, X (ω) = 3; If Omega is the wife, X (ω) = 4.

Random variables

Defining a random variable X is a measurable mapping (a measurable map) X:ω->r (a map that maps a set into a real number) so that any of Omega's elements Ω (that is, an event) is given a real number by X (ω).
Here, the measurable meaning is that for each X, there is {ω:x (ω) ≤x}∈a, where A is a Σ algebra, where the elements are measurable.
Therefore, the probability is a measure that acts on the set .

Distribution functions

Distribution functions (distribution function, also known as cumulation distribution function), are a mapping fx:r->[0,1].
FX (x) =p (x≤x), the distribution function FX maps the random variable corresponding to an event to a probability value of 0 to 1.

Application examples

So much so, how is random variables, probability distributions specifically linked to measurable mappings? We take the Bernoulli distribution as an example to introduce the implicit relationship.
Bernulli Distribution of PMF (probabilistic Mass Function) is

That is, when the probability of X=1 is P, when the probability of x=0 is 1-p.
Make sample space ω=[0,1], according to the Lebesgue measure, Pr ([A, b]) =b-a, wherein 0≤a<b≤1. Take a fixed p∈ (0,1), defined, when ω≤p, X (ω) = 1; When ω>p, X (ω) = 0.
So, PR (x=1) = PR (ω≤p) = PR ([0,p]) = p; Pr (x=0) =1-p.
Based on the above introduction, we can find that in the daily learning, in fact, it is omitted to map the collection to the real number of the implicit step.

Resources

Wiki: Lebesgue measure
Measurable space, measure space and σ algebra

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"Mathematics in machine learning" from sigma algebra, measure space to random variables