Posted on Categories:Uncertainty Quantification, 金融代写, 风险估值理论

# 金融代写|风险估值理论代写Uncertainty Quantification代考|ASEN6412 Elements of Functional Analysis

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## 金融代写|风险估值理论代写Uncertainty Quantification代考|Elements of Functional Analysis

Definition 1.1.1 A vector space $X$ over a scalar field $\mathbb{F}$ is an assemblage of elements, called vectors, that is closed under two algebraic operations, namely, vector addition and multiplication by scalars, that is, for any $u, v \in X$ and $\alpha \in \mathbb{F}$, we have $u+v \in X$ and $\alpha u \in X$. The algebraic operations satisfy the following:

1. For all $u, v \in X$, we have $u+v=v+u$.
2. For all $u, v, w \in X$, we have $u+(v+w)=(u+v)+w$.
3. There exists a vector, denoted by 0 , such that $u+0=u$, for any $u \in X$.
4. For any $u \in X$, there is a unique $-u \in X$ such that $u+(-u)=0$.
5. For any $\alpha, \beta \in \mathbb{F}$ and $u \in X$, we have $\alpha(\beta u)=(\alpha \beta) u$.
6. For the scalar 1 of $F$, we have $1 u=u$, for any $u \in X$.
7. For all $\alpha \in \mathbb{F}$ and $u, v \in X$, we have $\alpha(u+v)=\alpha u+\alpha v$.
8. For all $\alpha, \beta \in \mathbb{F}$ and $u \in X$, we have $(\alpha+\beta) u=\alpha u+\beta u$.

The vector space $X$ is called a real vector space, if $\mathbb{F}=\mathbb{R}$. The vector space $X$ is called a complex vector space, if $\mathbb{F}=\mathrm{C}$.

## 金融代写|风险估值理论代写Uncertainty Quantification代考|Fundamentals of Measure Theory and Integration

We now present an overview of the key ideas of measure theory and integration.
Definition 1.2.1 Let $X$ be a given set and $\mathcal{F}$ be a collection of subsets of $X$. The collection $\mathcal{F}$ is called a $\sigma$-algebra, if the following conditions hold:

1. $\emptyset \in \mathcal{F}$.
2. If $F \in \mathcal{F}$, then its complement $F^c:={x \in X \mid x \notin F} \in \mathcal{F}$.
3. If $F_j \in \mathcal{F}$, for $j=1,2, \ldots$, then $\bigcup_{j=1}^{\infty} F_j \in \mathcal{F}$.
If $X$ is a topological space, then the Borel $\sigma$-algebra, denoted by $\mathcal{B}(X)$, is the smallest $\sigma$-algebra containing all open subsets of $X$.

It is evident from the above definition that if $\mathcal{F}$ is a $\sigma$-algebra, then $X \in \mathcal{F}$, and if, $F, G \in \mathcal{F}$, then $F \backslash G=F \cap G^c \in \mathcal{F}$. Moreover, if $F_i \in \mathcal{F}$, for $i=1,2, \ldots$, then $\bigcap_{j=1}^{\infty} F_j \in \mathcal{F}$. The pair $(X, \mathcal{F})$ is termed as a measurable space and any set $F \in \mathcal{F}$ is called a measurable set.

Definition 1.2.2 A measure on a measurable space $(X, \mathcal{F})$ is a function $\mu: \mathcal{F} \rightarrow$ $[0,+\infty]$ satisfying the following conditions:

1. $\mu(\emptyset)=0$.
2. $\mu\left(\bigcup_{i=1}^{\infty} F_i\right)=\sum_{i=1}^{\infty} \mu\left(F_i\right)$, if $F_i \cap F_j=\emptyset$, for $i \neq j$.
The triple $(X, \mathcal{F}, \mu)$ is called a measure space. The measure space is called $\sigma$-finite, if $X=\cup_{j=1}^{\infty} F_j$, for $F_j \in \mathcal{F}$ with $\mu\left(F_j\right)<\infty$. A measure space $(X, \mathcal{F}, \mu)$ is called a complete measure space, if $F \in \mathcal{F}$ with $\mu(F)=0$ and $G \subset F$, then $G \in \mathcal{F}$. The set $F \in \mathcal{F}$ with $\mu(F)=0$ is called a null set.

## 金融代写|风险估值理论代写Uncertainty Quantification代考|Fundamentals of Measure Theory and Integration

㧴们现在概述则度论和集成的关键思想。

$\emptyset \in \mathcal{F}$.

$\mu(\emptyset)=0$.

$\mu\left(\bigcup_{i=1}^{\infty} F_i\right)=\sum_{i=1}^{\infty} \mu\left(F_i\right)$, 如果 $F_i \cap F_j=\emptyset$, 为了 $i \neq j$.

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