# 数学代写|概率论代写Probability theory代考|STAT4061

## 数学代写|概率论代写Probability theory代考|Product Integration and Fubini’s Theorem

In this section, let $\left(\Omega^{\prime}, L^{\prime}, I^{\prime}\right)$ and $\left(\Omega^{\prime \prime}, L^{\prime \prime}, I^{\prime \prime}\right)$ be two arbitrary but fixed complete integration spaces. Let $\Omega \equiv \Omega^{\prime} \times \Omega^{\prime \prime}$ denote the product set. We will construct the product integration space and embed the given integration spaces in it. The definitions and results can easily be generalized to more than two given integration spaces.

Definition 4.10.1. Direct product of functions. Let $X^{\prime}, X^{\prime \prime}$ be arbitrary members of $L^{\prime}, L^{\prime \prime}$, respectively. Define the function $X^{\prime} \otimes X^{\prime \prime}: \Omega \rightarrow R$ by $\operatorname{domain}\left(X^{\prime} \otimes\right.$ $\left.X^{\prime \prime}\right) \equiv \operatorname{domain}\left(X^{\prime}\right) \times \operatorname{domain}\left(X^{\prime \prime}\right)$ and by $\left(X^{\prime} \otimes X^{\prime \prime}\right)\left(\omega^{\prime}, \omega^{\prime \prime}\right) \equiv X^{\prime}\left(\omega^{\prime}\right) X^{\prime \prime}\left(\omega^{\prime \prime}\right)$ for each $\omega \in \Omega$. The function $X^{\prime} \otimes X^{\prime \prime}$ is then called the direct product of the functions $X^{\prime}$ and $X^{\prime \prime}$. When the risk of confusion is low, we will write $X^{\prime} \otimes X^{\prime \prime}$ and $X^{\prime} X^{\prime \prime}$ interchangeably.

Definition 4.10.2. Simple functions. Let $n, m \geq 1$ be arbitrary. Let $X_1^{\prime}, \ldots, X_n^{\prime} \in$ $L^{\prime}$ be mutually exclusive indicators, and let $X_1^{\prime \prime}, \ldots, X_m^{\prime \prime} \in L^{\prime \prime}$ be mutually exclusive indicators. For each $i=1, \ldots, n$ and $j=1, \ldots, m$, let $c_{i, j} \in R$ be arbitrary. Then the real-valued function
$$X=\sum_{i=1}^n \sum_{j=1}^m c_{i, j} X_i^{\prime} X_j^{\prime \prime}$$
is called a simple function relative to $L^{\prime}, L^{\prime \prime}$. Let $L_0$ denote the set of simple functions on $\Omega^{\prime} \times \Omega^{\prime \prime}$. Two simple functions are said to be equal if they have the same domain and the same values on the common domain. In other words, equality in $L_0$ is the set-theoretic equality:
$$I(X)=\sum_{i=1}^n \sum_{j=1}^m c_{i, j} I^{\prime}\left(X_i^{\prime}\right) I^{\prime \prime}\left(X_j^{\prime \prime}\right)$$
Lemma 4.10.3. Simple functions constitute a linear space. As in Definition $4.10 .2$, let $L_0$ be the set of simple functions on $\Omega^{\prime} \times \Omega^{\prime \prime}$ relative to $L^{\prime}, L^{\prime \prime}$. Then the following conditions hold:

1. If $X \in L_0$, then $|X|, a \wedge X \in L_0$ for each $a>0$.
2. $L_0$ is a linear space.
3. The function I on $L_0$ is linear:

## 数学代写|概率论代写Probability theory代考|Random Variable

Definition 5.1.1. Probability space and r.v.’s. Henceforth, unless otherwise specified, $(\Omega, L, E)$ will denote a probability integration space, i.e., a complete integration space in which the constant function 1 is integrable with $E 1=1$. Then $(\Omega, L, E)$ will simply be called a probability space. The integration $E$ will be called an expectation, and the integral $E X$ of each $X \in L$ will be called the expected value of $X$.

A measurable function $X$ on $(\Omega, L, E)$ with values in a complete metric space $(S, d)$ is called a random variable, or r.v. for abbreviation. Two r.v.’s are considered equal if they have equal values on a full subset of $\Omega$. A real-valued measurable function $X$ on $(\Omega, L, E)$ is then called a real random variable, or r.r.v. for abbreviation. An integrable real-valued function $X$ is called an integrable real random variable, its integral $E X$ called its expected value.

A measurable set is sometimes called an event. It is then integrable because $1_A \leq 1$, and its measure $\mu(A)$ is called its probability and denoted by $P(A)$ or $P A$. The function $P$ on the set of measurable sets is called the probability function corresponding to the expectation $E$. Sometimes we will write $E(A)$ for $P(A)$. The set $\Omega$ is called the sample space, and a point $\omega \in \Omega$ is called a sample or an outcome. If an outcome $\omega$ belongs to an event $A$, the event $A$ is said to occur for $\omega$, and $\omega$ is said to realize $A$.

The terms “almost surely,” “almost sure,” and the abbreviation “a.s.” will stand for “almost everywhere” or its abbreviation “a.e.” Henceforth, unless otherwise specified, equality of r.v.’s and equality of events will mean a.s. equality, and the term “complement” for events will stand for “measure-theoretic complement.” If $X$ is an integrable r.r.v. and $A, B, \ldots$ are events, we will sometimes write $E(X ; A, B, \ldots)$ for $E X 1_{A B} \ldots$.

Let $X \in L$ be arbitrary. We will sometimes use the more suggestive notation
$$\int E(d \omega) X(\omega) \equiv E X,$$
where $\omega$ is a dummy variable. For example, if $Y \in L \otimes L \otimes L$, we can define a function $Z \in L \otimes L$ by the formula
$$Z\left(\omega_1, \omega_3\right) \equiv \int E\left(d \omega_2\right) Y\left(\omega_1, \omega_2, \omega_3\right) \equiv E Y\left(\omega_1, \cdot, \omega_3\right)$$
for each $\left(\omega_1, \omega_3\right) \in \Omega^2$ for which the right-hand side is defined.

## 数学代写|概率论代写Probability theory代考|Product Integration and Fubini’s Theorem

$$X=\sum_{i=1}^n \sum_{j=1}^m c_{i, j} X_i^{\prime} X_j^{\prime \prime}$$

$$I(X)=\sum_{i=1}^n \sum_{j=1}^m c_{i, j} I^{\prime}\left(X_i^{\prime}\right) I^{\prime \prime}\left(X_j^{\prime \prime}\right)$$

1. 如果 $X \in L_0$ ，然后 $|X|, a \wedge X \in L_0$ 对于每个 $a>0$.
2. $L_0$ 是一个线性空间。
3. 我的功能 $L_0$ 是线性的:

## 数学代写|概率论代写Probability theory代考|Random Variable

$$\int E(d \omega) X(\omega) \equiv E X$$

$$Z\left(\omega_1, \omega_3\right) \equiv \int E\left(d \omega_2\right) Y\left(\omega_1, \omega_2, \omega_3\right) \equiv E Y\left(\omega_1, \cdot, \omega_3\right)$$

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