### 统计代写|概率论作业代写Probability and Statistics代考5CCM241A|Algebras, sigma-algebras and measurable spaces

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## 统计代写|概率论作业代写Probability and Statistics代考5CCM241A|Algebras and sigma-algebras

As we have already noted, algebras and $\sigma$-algebras are constituent elements in the construction of probability spaces. In this subsection we will first give some important examples of algebras and $\sigma$-algebras. Then we will prove a number of statements that will be used further.
Let $\Omega={\omega}$ is some sample space. Then the sets of systems
$$\mathcal{F}{}={\varnothing, \Omega}, \quad \mathcal{F}^{}={A: A \subseteq \Omega}$$
are algebras and $\sigma$-algebras.
By definition, $\mathcal{F}^{}$ contains all subsets of the sample space $\Omega$ and is the «richest» $\sigma$-algebra, and $\mathcal{F}{}$ is the «poorest» $\sigma$-algebra.
If $A \subseteq \Omega$, then the system
$$\mathcal{F}{\mathrm{A}}={\varnothing, A, \bar{A}, \Omega}$$ is also a $\sigma$-algebra (it is called the $\sigma$-algebra, generated by the event $A$ ). If $D=\left{D{1}, D_{2}, \ldots\right}$ is an countable partition of $\Omega$ (i.e. $D_{i} \subseteq \Omega, D_{i} \neq \varnothing$, $\left.D_{i} D_{j}=\varnothing(i \neq j), \sum_{i} D_{i}=\Omega\right)$, then the system
$$\alpha(D)=\left{\sum_{j=1}^{n} D_{i j}, i_{j} \neq i_{l}(j \neq l), n<\infty\right}$$
is an algebra and this algebra is called an algebra generated by the partition $D$.

## 统计代写|概率论作业代写Probability and Statistics代考5CCM241A|The theorem on the continuation of probability

Let’s return to the definition of a probability space.
Let a triple $(\Omega, \mathcal{A}, P)$ form a probability space in the broad sense ( $\mathcal{A}$ is algebra).
As we have seen, we can associate with the algebra $\mathcal{A}$ the smallest $\sigma$-algebra $\sigma(\mathcal{A})$ containing $\mathcal{A}(\sigma(\mathcal{A})$ is the smallest $\sigma$-algebra generated by the algebra $\mathcal{A})$.

The following question is of considerable interest for probability theory: does the probability measure $P$ on $\mathcal{A}$ determine a probability measure on $\mathcal{F}=\sigma(\mathcal{A})$ and is this uniquely true?

In other words, is it sufficient to define the probability $P$ only on some algebra $\mathcal{A}$ that generates $\mathcal{F}$ (i.e. to construct a probability space $(\Omega, \mathcal{A}, P)$ in the broad sense with $\sigma(\mathcal{A})=\mathcal{F})$ for the construction a probability space $(\Omega, \mathcal{F}, P)$ ?

The answer to this question is given by the following theorem of Carathedori (theorem on the extension of probability (probability measure)).

Theorem (the theorem of Carathedori on the extension of probability). Let $(\Omega, \mathcal{A}, P)$ be an extended probability space.

Then on $\mathcal{F}=\sigma(\mathcal{A})$ there is a unique probabilistic measure $Q$, such that $Q(A)=P(A)$ for all $A \in \mathcal{A}$.

Here we do not give a proof of this theorem. The proof of this theorem adapted to the probability measure is given in [11] (see [11], pp. 308-314).

Any extended probability space $(\Omega, \mathcal{A}, P)$ automatically defines the probability space $(\Omega, \mathcal{F}, P)$, where $\mathcal{F}=\sigma(\mathcal{A})$ is the smallest $\sigma$-algebra containing the algebra.

## 统计代写|概率论作业代写Probability and Statistics代考5CCM241A|The most important examples of measurable spaces

Borel $\sigma$-algebra $\beta(R)$. Let $R=(-\infty,+\infty)$ is a real number scale, $(a, b]={x \in R: a<x \leq b}$ for all $-\infty \leq a<b<+\infty$. We agree to understand such an interval $(a, \infty]$ as the interval $(a, \infty)$. (This agreement is necessary in order for the complement to the interval $(-\infty, b]$ to be an interval of the same kind-open to the left and closed to the right).
Let’s define the set system $\mathcal{A}$ as follows:
$$\mathcal{A}=\left{A: A=\sum_{i=1}^{n}\left(a_{i}, b_{i}\right], n<\infty\right} .$$
A system $\mathcal{A}$ with an empty set $\varnothing$ included in it is an algebra, but is not a $\sigma$-algebra (for example, if $A_{n}=(0,1-1 / n] \in \mathcal{A}$, then $\bigcup_{n=1}^{\infty} A_{n}=(0,1) \notin \mathcal{A}, n=1,2, \ldots$ )
The smallest $\sigma$-algebra $\sigma(\mathcal{A})$, containing $\mathcal{A}$, is called a Borel $\sigma$-algebra on the number scale, and the elements of the Borel $\sigma$-algebra are called the Borel sets.

Everywhere further, according to the tradition, a $\sigma$-algebra defined in this way will be denoted by $\beta(R)$ (or $\beta$, or $\beta_{1}$ ).

If we introduce the system of intervals $J={I: I=(a, b]}$ and denote by $\sigma(J)$ the smallest $\sigma$-algebra which contains $J$, then it is not difficult to verify that $\sigma(J)=\beta(R)$. In other words, one can come to the Borel $\sigma$-algebra from the system $J$, without of reference to the algebra $\mathcal{A}$, because $\sigma(J)=\sigma(\alpha(J))(\alpha(J)$ is the smallest algebra which contains $J$ ).
Note that
$$(a, b)=\bigcup_{n=1}^{\infty}\left(a, b-\frac{1}{n}\right], a<b ; \quad[a, b]=\bigcap_{n=1}^{\infty}\left(a-\frac{1}{n}, b\right], a<b ;{a}=\bigcap_{n=1}^{\infty}\left(a-\frac{1}{n}, a\right]$$

These ratios show that in the Borel $\sigma$-algebra $\beta(R)$, in addition to the intervals of the form $(a, b]$, there are one-point sets ${a}$ and all intervals of the forms $(a, b),[a, b],[a, b),(-\infty, b),(-\infty, b],(a, \infty)$.

From what has been said, we conclude that we can construct a Borel $\sigma$-algebra $\beta(R)$ based not only on the intervals of the form $(a, b]$, but also on any of the forms of the last six intervals.

Thus, the Borel $\sigma$-algebra $\beta(R)$ on the number scale is the smallest $\sigma$-algebra containing all possible intervals on the number scale.

Roughly speaking, a Borel $\sigma$-algebra can be imagined as a collection of sets obtained from intervals by means of a countable number of operations of union, intersection, and taking of complements.

Measurable space $(R, \beta(R))$ will be indicated sometimes by $(R, \beta)$, sometimes by $\left(R_{1}, \beta_{1}\right)$.

## 统计代写|概率论作业代写Probability and Statistics代考5CCM241A|Algebras and sigma-algebras

F=∅,Ω,F=一种:一种⊆Ω

F一种=∅,一种,一种¯,Ω也是一个σ-代数（它被称为σ-代数，由事件产生一种）。如果D=\left{D{1}, D_{2}, \ldots\right}D=\left{D{1}, D_{2}, \ldots\right}是一个可数分区Ω（IED一世⊆Ω,D一世≠∅, D一世Dj=∅(一世≠j),∑一世D一世=Ω), 那么系统
\alpha(D)=\left{\sum_{j=1}^{n} D_{i j}, i_{j} \neq i_{l}(j \neq l), n<\infty\right}\alpha(D)=\left{\sum_{j=1}^{n} D_{i j}, i_{j} \neq i_{l}(j \neq l), n<\infty\right}

## 统计代写|概率论作业代写Probability and Statistics代考5CCM241A|The most important examples of measurable spaces

\mathcal{A}=\left{A: A=\sum_{i=1}^{n}\left(a_{i}, b_{i}\right], n<\infty\right} 。\mathcal{A}=\left{A: A=\sum_{i=1}^{n}\left(a_{i}, b_{i}\right], n<\infty\right} 。

(一种,b)=⋃n=1∞(一种,b−1n],一种<b;[一种,b]=⋂n=1∞(一种−1n,b],一种<b;一种=⋂n=1∞(一种−1n,一种]

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