## 数学代写|概率论代写Probability theory代考|Complete Extension of Integration

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## 数学代写|概率论代写Probability theory代考|Complete Extension of Integration

A common and useful integration space is $(S, C(S, d), I)$, where $(S, d)$ is a locally compact metric space and where $I$ is an integration in the sense of Definition 4.2.1. However, the family $C(S, d)$ of continuous functions with compact support is too narrow to hold all the interesting integrable functions in applications. For example, in the case where $(S, d)$ is the unit interval $[0,1]$ with the Euclidean metric and where $I$ is the Lebesgue integral, it will be important to be able to integrate simple step functions like the indicator $1_{\left[2^{-1}, 1\right]}$. For that reason, we will expand the family $C(S, d)$ to a family of integrable functions that includes functions which need not be continuous, and that includes an abundance of indicators.

More generally, given an integration space $(\Omega, L, I)$, we will expand the family $L$ to a larger family $L_1$ and extend the integration $I$ to $L_1$. We will do so by summing certain series of small pieces in $L$, in a sense to be made precise presently. This is analogous to of the expansion of the set of rational numbers to the set of real numbers by representing a real number as the sum of a convergent series of rational numbers.

Definition 4.4.1. Integrable functions and complete extension of an integration space. Let $(\Omega, L, I)$ be an arbitrary integration space. Recall that each function $X$ on $\Omega$ is required to have a nonempty $\operatorname{domain}(X)$.

A function $X$ on $\Omega$ is called an integrable function if there exists a sequence $\left(X_n\right){n=1,2, \ldots .}$ in $L$ satisfying the following two conditions: (i) $\sum{i=1}^{\infty} I\left|X_i\right|<\infty$.
(ii) For each $\omega \in \bigcap_{i=1}^{\infty} \operatorname{domain}\left(X_i\right)$ such that $\sum_{i=1}^{\infty}\left|X_i(\omega)\right|<\infty$, we have $\omega \in \operatorname{domain}(X)$ and
$$X(\omega)=\sum_{i=1}^{\infty} X_i(\omega) .$$

## 数学代写|概率论代写Probability theory代考|Integrable Set

To model an event in an experiment of chance that may or may not occur depending on the outcome, we can use a function of the outcome with only two possible values, 1 or 0 . Equivalently, we can specify the subset of those outcomes that realize the event. We make these notions precise in the present section.

Definition 4.5.1. Indicators and mutually exclusive subsets. A function $X$ on a set $\Omega$ with only two possible values, 1 or 0 , is called an indicator.

Subsets $A_1, \ldots, A_n$ of a set $\Omega$ are said to be mutually exclusive if $A_i A_j=\phi$ for each $i, j=1, \ldots, n$ with $i \neq j$. Indicators $X_1, \ldots, X_n$ are said to be mutually exclusive if the sets $\left{\omega \in \operatorname{domain}\left(X_i\right): X_i(\omega)=1\right}(i=1, \ldots, n)$ are mutually exclusive.

In the remainder of this section, let $(\Omega, L, I)$ be a complete integration space. Recall that an integrable function need not be defined everywhere. However, such functions are defined almost everywhere in the sense of the next definition.

Definition 4.5.2. Full set, and conditions holding almost everywhere. A subset $D$ of $\Omega$ is called a full set if $D \supset \operatorname{domain}(X)$ for some integrable function $X \in L=L_1$. By Definition 4.4.1, domain $(X)$ is nonempty for each integrable function $X \in L_1$. Hence each full set $D$ is nonempty.

Two integrable functions $Y, Z \in L_1$ are said to be equal almost everywhere (or $Y=Z$ a.e. in symbols) if there exists a full set $D \subset \operatorname{domain}(Y) \cap \operatorname{domain}(Z)$ such that $Y=Z$ on $D$.

More generally, a condition about a general element $\omega$ of $\Omega$ is said to hold almost everywhere (a.e. for short) if it holds for each $\omega$ in a full set $D$. 다이

For example, according to the terminology established in the Introduction of this book, the statement $Y \leq Z$ means that for each $\omega \in \Omega$ we have (i’) $\omega \in$ $\operatorname{domain}(Y)$ iff $\omega \in \operatorname{domain}(Z)$, and (ii’) $Y(\omega) \leq Z(\omega)$ if $\omega \in \operatorname{domain}(Y)$. Hence the statement $Y \leq Z$ a.e. means that there exists some full set $D$ such that, for each $\omega \in D$, Conditions (i’) and (ii’) hold.
A similar argument holds when $\leq$ is replaced by $\geq$ or by $=$.
If $A, B$ are subsets of $\Omega$, then $A \subset B$ a.e. iff $A D \subset B D$ for some full set $D$.

# 概率论代考

## 数学代写|概率论代写Probability theory代考|Complete Extension of Integration

(ii) 对于每个 $\omega \in \bigcap_{i=1}^{\infty} \operatorname{domain}\left(X_i\right)$ 这样 $\sum_{i=1}^{\infty}\left|X_i(\omega)\right|<\infty$ ，我们有 $\omega \in \operatorname{domain}(X)$ 和
$$X(\omega)=\sum_{i=1}^{\infty} X_i(\omega) .$$

## 数学代写|概率论代写Probability theory代考|Integrable Set

$\omega \in \operatorname{domain}(Y)$. 因此声明 $Y \leq Z$ ae 表示存在一些完 整的集合 $D$ 这样，对于每个 $\omega \in D$ ，条件 (i’) 和 (ii’) 成 立。

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## MATLAB代写

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