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• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|概率论代写Probability theory代考|Exchanging Integral and Differentiation

We study how properties such as continuity and differentiability of functions of two variables behave under integration with respect to one of the variables.

Theorem 6.27 (Continuity lemma) Let $(E, d)$ be a metric space, $x_0 \in E$ and let $f: \Omega \times E \rightarrow \mathbb{R}$ be a map with the following properties.
(i) For any $x \in E$, the map $\omega \mapsto f(\omega, x)$ is in $\mathcal{L}^1(\mu)$.
(ii) For almost all $\omega \in \Omega$, the map $x \mapsto f(\omega, x)$ is continuous at the point $x_0$.
(iii) There is a map $h \in \mathcal{L}^1(\mu), h \geq 0$, such that $|f(\cdot, x)| \leq h \quad \mu$-a.e.for all $x \in E$.
Then the map $F: E \rightarrow \mathbb{R}, x \mapsto \int f(\omega, x) \mu(d \omega)$ is continuous at $x_0$.
Proof Let $\left(x_n\right){n \in \mathbb{N}}$ be a sequence in $E$ with $\lim {n \rightarrow \infty} x_n=x_0$. Define $f_n=f\left(\cdot, x_n\right)$.
By assumption, $\left|f_n\right| \leq h$ and $f_n \stackrel{n \rightarrow \infty}{\longrightarrow} f\left(\cdot, x_0\right)$ almost everywhere. By the dominated convergence theorem (Corollary $6.26$ ), we get
$$F\left(x_n\right)=\int f_n d \mu \stackrel{n \rightarrow \infty}{\longrightarrow} \int f\left(\cdot, x_0\right) d \mu=F\left(x_0\right) .$$
Hence $F$ is continuous at $x_0$.
Reflection Find an example that shows that condition (iii) in Theorem $6.27$ cannot simply be dropped.

Theorem 6.28 (Differentiation lemma) Let $I \subset \mathbb{R}$ be a nontrivial open interval and let $f: \Omega \times I \rightarrow \mathbb{R}$ be a map with the following properties.
(i) For any $x \in E$, the map $\omega \mapsto f(\omega, x)$ is in $\mathcal{L}^1(\mu)$.
(ii) For almost all $\omega \in \Omega$, the map $I \rightarrow \mathbb{R}, x \mapsto f(\omega, x)$ is differentiable with derivative $f^{\prime}$.
(iii) There is a map $h \in \mathcal{L}^1(\mu), h \geq 0$, such that $\left|f^{\prime}(\cdot, x)\right| \leq h \quad \mu$-a.e. for all $x \in I$.

Then, for any $x \in I, f^{\prime}(\cdot, x) \in \mathcal{L}^1(\mu)$ and the function $F: x \mapsto \int f(\omega, x) \mu(d \omega)$ is differentiable with derivative
$$F^{\prime}(x)=\int f^{\prime}(\omega, x) \mu(d \omega) .$$

## 统计代写|概率论代写Probability theory代考|$L^p$-Spaces and the Radon-Nikodym Theorem

We always assume that $(\Omega, \mathcal{A}, \mu)$ is a $\sigma$-finite measure space. In Definition 4.16, for measurable $f: \Omega \rightarrow \overline{\mathbb{R}}$, we defined
$$|f|_p:=\left(\int|f|^p d \mu\right)^{1 / p} \quad \text { for } p \in[1, \infty)$$
and
$$|f|_{\infty}:=\inf {K \geq 0: \mu(|f|>K)=0}$$
Further, we defined the spaces of functions where these expressions are finite: $\mathcal{L}^p(\Omega, \mathcal{A}, \mu)=\mathcal{L}^p(\mathcal{A}, \mu)=\mathcal{L}^p(\mu)=\left{f: \Omega \rightarrow \overline{\mathbb{R}}\right.$ measurable and $\left.|f|_p<\infty\right}$
We saw that $|\cdot|_1$ is a seminorm on $\mathcal{L}^1(\mu)$. Here our first goal is to change $|\cdot|_p$ into a proper norm for all $p \in[1, \infty]$. Apart from the fact that we still have to show the triangle inequality, to this end, we have to change the space a little bit since we only have
$$|f-g|_p=0 \Longleftrightarrow f=g \quad \mu \text {-a.e. }$$
For a proper norm (that is, not only a seminorm), the left-hand side has to imply equality (not only a.e.) of $f$ and $g$. Hence we now consider $f$ and $g$ as equivalent if $f=g$ almost everywhere. Thus let
$\mathcal{N}={f$ is measurable and $f=0 \quad \mu=$ a.e. $}$.
For any $p \in[1, \infty], \mathcal{N}$ is a subvector space of $\mathcal{L}^p(\mu)$. Thus formally we can build the factor space. This is the standard procedure in order to change a seminorm into a proper norm.

# 概率论代考

## 统计代写|概率论代写概率论代考|交换积分与微分

(i)对于任何 $x \in E$，地图 $\omega \mapsto f(\omega, x)$ 在 $\mathcal{L}^1(\mu)$.
(ii)几乎所有的 $\omega \in \Omega$，地图 $x \mapsto f(\omega, x)$ 在点上是连续的吗 $x_0$
(iii)有一张地图 $h \in \mathcal{L}^1(\mu), h \geq 0$，以致于 $|f(\cdot, x)| \leq h \quad \mu$-a。e。for all $x \in E$.

$$F\left(x_n\right)=\int f_n d \mu \stackrel{n \rightarrow \infty}{\longrightarrow} \int f\left(\cdot, x_0\right) d \mu=F\left(x_0\right) .$$

(i)对于任何$x \in E$，映射$\omega \mapsto f(\omega, x)$在$\mathcal{L}^1(\mu)$中。
(ii)对于几乎所有$\omega \in \Omega$，映射$I \rightarrow \mathbb{R}, x \mapsto f(\omega, x)$可与导数$f^{\prime}$可微。
(iii)存在一个映射$h \in \mathcal{L}^1(\mu), h \geq 0$，使得$\left|f^{\prime}(\cdot, x)\right| \leq h \quad \mu$对于所有$x \in I$。

.
(i)对于任何，映射在中。
(ii)对于几乎所有，映射与导数可微

$$F^{\prime}(x)=\int f^{\prime}(\omega, x) \mu(d \omega) .$$

## 统计代写|概率论代写概率论代考| $L^p$ -spaces and the Radon-Nikodym定理

$$|f|p:=\left(\int|f|^p d \mu\right)^{1 / p} \quad \text { for } p \in[1, \infty)$$

$$|f|{\infty}:=\inf {K \geq 0: \mu(|f|>K)=0}$$

$$|f-g|_p=0 \Longleftrightarrow f=g \quad \mu \text {-a.e. }$$

$\mathcal{N}={f$是可测量的，$f=0 \quad \mu=$ a.e. $}$ .

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