## 数学代写|概率论代写Probability theory代考|Measurable Function and Measurable Set

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## 数学代写|概率论代写Probability theory代考|Measurable Function and Measurable Set

In this section, let $(\Omega, L, I)$ be a complete integration space, and let $(S, d)$ be a complete metric space with a fixed reference point $x_{\circ} \in S$. In the case where $S=R$, it is understood that $d$ is the Euclidean metric and that $x_0=0$.

Recall that, as an abbreviation, we write $A B \equiv A \cap B$ for subsets $A$ and $B$ of $\Omega$. Recall from the notations and conventions described in the Introduction that if $X$ is a real-valued function on $\Omega$ and if $t \in R$, then we use the abbreviation $(t \leq X)$ for the subset ${\omega \in \operatorname{domain}(X): t \leq X(\omega)}$. Similarly, ” $\leq$ ” may be replaced by ” $\geq$,” ” $<$,” or “=” As usual, we write $a_b$ interchangeably with $a(b)$ to lessen the burden on subscripts.

Recall that $\mu A \equiv \mu(A)$ stands for the measure of an integrable subset $A$ of $\Omega$. We next generalize the theory of real-valued measurable functions in [Bishop and Bridges 1985] to a theory of measurable functions with values in the complete metric space $(S, d)$. Recall that $C_{u b}(S)$ stands for the space of bounded and uniformly continuous real-valued functions on $S$.

Definition 4.8.1. Measurable function. A function $X$ from $(\Omega, L, I)$ to the complete metric space $(S, d)$ is called a measurable function if
(i) for each integrable set $A$ and each $f \in C_{u b}(S)$, we have $f(X) 1_A \in L$, and
(ii) for each integrable set $A$, there exists a countable subset $J$ of $R$ such that $\left.\mu\left(d\left(x_0, X\right)>a\right) A\right) \rightarrow 0$ as $a \rightarrow \infty$ while $a$ remains in the metric complement $J_c$ of $J$

In the special case where the constant function 1 is integrable, then Conditions (i) and (ii) reduce to $\left(\mathrm{i}^{\prime}\right) f(X) \in L$ and $\left(\mathrm{ii}^{\prime}\right) \mu\left(d\left(x_{\circ}, X\right)>a\right) \rightarrow 0$ as $a \rightarrow \infty$ while $a \in J_c$.

It is obvious that if Condition (ii) holds for one point $x_0 \in S$, then it holds for any point $x_{\circ}^{\prime} \in S$. The next lemma shows that, given Condition (i) and given an arbitrary integrable set $A$, the measure in Condition (ii) is welldefined for all but countably many $a \in R$. Hence Condition (ii) makes sense.

## 数学代写|概率论代写Probability theory代考|Convergence of Measurable Functions

In this section, let $(\Omega, L, I)$ be a complete integration space, and let $(S, d)$ be a complete metric space, with a fixed reference point $x_{\circ} \in S$. In the case where $S=R$, it is understood that $d$ is the Euclidean metric and that $x_0=0$. We will introduce several notions of convergence of measurable functions on $(\Omega, L, I)$ with values in $(S, d)$. We will sometimes write $\left(a_i\right)$ for short for a given sequence $\left(a_i\right){i=1,2, \ldots}$ Recall the following definition. Definition 4.9.1. Limit of a sequence of functions. If $\left(Y_i\right){i=1,2, \ldots}$ is a sequence of functions from a set $\Omega^{\prime}$ to the metric space $(S, d)$, and if the set
$$D \equiv\left{\omega \in \bigcap_{i=1}^{\infty} \operatorname{domain}\left(Y_i\right): \lim {i \rightarrow \infty} Y_i(\omega) \text { exists in }(S, d)\right}$$ is nonempty, then the function $\lim {i \rightarrow \infty} Y_i$ is defined by domain $\left(\lim {i \rightarrow \infty} Y_i\right) \equiv D$ and by $\left(\lim {i \rightarrow \infty} Y_i\right)(\omega) \equiv \lim _{i \rightarrow \infty} Y_i(\omega)$ for each $\omega \in D$.

Definition 4.9.2. Convergence in measure, a.u., a.e., and in $L_1$. For each $n \geq 1$, let $X, X_n$ be functions on the complete integration space $(\Omega, L, I)$, with values in the complete metric space $(S, d)$.

1. The sequence $\left(X_n\right)$ is said to converge to $X$ uniformly on a subset $A$ of $\Omega$ if for each $\varepsilon>0$, there exists $p \geq 1$ so large that $A \subset \bigcap_{n=p}^{\infty}\left(d\left(X_n, X\right) \leq \varepsilon\right)$.
2. The sequence $\left(X_n\right)$ is said to converge to $X$ almost uniformly (a.u.) if for each integrable set $A$ and for each $\varepsilon>0$, there exists an integrable set $B$ with $\mu(B)<\varepsilon$ such that $X_n$ converges to $X$ uniformly on $A B^c$.
3. The sequence $\left(X_n\right)$ is said to converge to $X$ in measure if for each integrable set $A$ and for each $\varepsilon>0$, there exists $p \geq 1$ so large that for each $n \geq p$, there exists an integrable set $B_n$ with $\mu\left(B_n\right)<\varepsilon$ and $A B_n^c \subset\left(d\left(X_n, X\right) \leq \varepsilon\right)$.
4. The sequence $\left(X_n\right)$ is said to be Cauchy in measure if for each integrable set $A$ and for each $\varepsilon>0$, there exists $p \geq 1$ so large that for each $m, n \geq p$ there exists an integrable set $B_{m, n}$ with $\mu\left(B_{m, n}\right)<\varepsilon$ and $A B_{m, n}^c \subset\left(d\left(X_n, X_m\right) \leq \varepsilon\right)$.
5. Suppose $S=R$ and $X, X_n \in L$ for $n \geq 1$. The sequence $\left(X_n\right)$ is said to converge to $X$ in $L_1$ if $I\left|X_n-X\right| \rightarrow 0$.

# 概率论代考

## 数学代写|概率论代写Probability theory代考|Convergence of Measurable Functions

4.9.1。函数序列的限制。如果 $\left(Y_i\right) i=1,2, \ldots$ 是一组 函数的序列 $\Omega^{\prime}$ 到度量空间 $(S, d)$ ，如果集合

$\left(\lim i \rightarrow \infty Y_i\right) \equiv D$ 并通过
$\left(\lim i \rightarrow \infty Y_i\right)(\omega) \equiv \lim _{i \rightarrow \infty} Y_i(\omega)$ 每个 $\omega \in D$.

$(\Omega, L, I)$, 具有完整度量空间中的值 $(S, d)$.

1. 序列 $\left(X_n\right)$ 据说收敛于 $X$ 在子集上一致 $A$ 的 $\Omega$ 如果 对于每个 $\varepsilon>0$ ， 那里存在 $p \geq 1$ 大到 $A \subset \bigcap_{n=p}^{\infty}\left(d\left(X_n, X\right) \leq \varepsilon\right)$.
2. 序列 $\left(X_n\right)$ 据说收敛于 $X$ 几乎一致 (au) 如果对于 每个可积集 $A$ 并为每个 $\varepsilon>0$, 存在可积集 $B$ 和 $\mu(B)<\varepsilon$ 这样 $X_n$ 收敛于 $X$ 统一上 $A B^c$.
3. 序列 $\left(X_n\right)$ 据说收敛于 $X$ 衡量每个可积集 $A$ 并为每 个 $\varepsilon>0$ ， 那里存在 $p \geq 1$ 大到每个 $n \geq p$, 存在 可积集 $B_n$ 和 $\mu\left(B_n\right)<\varepsilon$ 和 $A B_n^c \subset\left(d\left(X_n, X\right) \leq \varepsilon\right)$.
4. 序列 $\left(X_n\right)$ 如果对于每个可积集，则称其为柯西测 度 $A$ 并为每个 $\varepsilon>0$ ，那里存在 $p \geq 1$ 大到每个 $m, n \geq p$ 存在可积集 $B_{m, n}$ 和 $\mu\left(B_{m, n}\right)<\varepsilon$ 和 $A B_{m, n}^c \subset\left(d\left(X_n, X_m\right) \leq \varepsilon\right)$.
5. 认为 $S=R$ 和 $X, X_n \in L$ 为了 $n \geq 1$. 序列 $\left(X_n\right)$ 据说收敛于 $X$ 在 $L_1$ 如果 $I\left|X_n-X\right| \rightarrow 0$.

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