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# 数学代写|勒贝格积分代写Lebesgue Integration代考|BSMA2003 Measurable Functions

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## 数学代写|勒贝格积分代写Lebesgue Integration代考|Measurable Functions

In this chapter we want to define the Lebesgue integral in a fashion which is analogous to our definitions of regulated integral and Riemann integral from Chapter 1. The difference is that we will no longer use step functions to approximate a function we want to integrate, but instead will use a much more general class called simple functions.
Definition 3.1.1. (Characteristic function). If $A \subset[0,1]$, its characteristic function $\mathfrak{X}A(x)$ (sometimes called the indicator function) is defined by $$\mathfrak{X}_A(x)= \begin{cases}1, & \text { if } x \in A ; \ 0, & \text { otherwise } .\end{cases}$$ Definition 3.1.2. (Measurable partition). A finite measurable partition of $[0,1]$ is a collection $\left{A_i\right}{i=1}^n$ of measurable subsets which are pairwise disjoint and whose union is $[0,1]$.

We can now define simple functions. Like step functions these functions have only finitely many values, but unlike step functions the set on which a simple function assumes a given value is no longer an interval. Instead, a simple function is constant on each subset of a finite measurable partition of $[0,1]$.

## 数学代写|勒贝格积分代写Lebesgue Integration代考|The Lebesgue Integral of Bounded Functions

In this section we want to define the Lebesgue integral and characterize the bounded integrable functions. In the case of the regulated integral, the integrable functions are the uniform limits of step functions. In the case of the Riemann integral a function $f$ is integrable if the infimum of the integrals of step functions larger than $f$ equals the supremum of the integrals of step functions less than $f$. It is natural to alter both of these definitions, replacing step functions with simple functions. It turns out that when we do this for bounded functions we get the same class of integrable functions whether we use the analog of regulated integral or the analog of Riemann integral. Moreover, this class is precisely the bounded measurable functions!

Theorem 3.2.1. If $f:[0,1] \rightarrow \mathbb{R}$ is a bounded function, then the following are equivalent:
(1) The function $f$ is Lebesgue measurable.
(2) There is a sequence of simple functions $\left{f_n\right}_{n=1}^{\infty}$ which converges uniformly to $f$.
(3) If $\mathcal{U}\mu(f)$ denotes the set of all simple functions $u(x)$ such that $f(x) \leq u(x)$ for all $x$ and if $\mathcal{L}\mu(f)$ denotes the set of all simple functions $v(x)$ such that $v(x) \leq f(x)$ for all $x$, then
$$\sup {v \in \mathcal{L}\mu(f)}\left{\int v d \mu\right}=\inf {u \in \mathcal{U}\mu(f)}\left{\int u d \mu\right}$$

## 数学代写|勒贝格积分代写|勒贝格积分代考|可测量函数

$mathfrak{X}_A(x)={1, \quad$ 如果$x\在A中; 0, \quad$ 否则。

## 数学代写勒贝格积分代写Lebesgue Integration代写|The Lebesgue Integral of Bounded Functions

(1) 函数$f$是Lebesgue可测的。
(2) 存在一个简单的函数序列 缺少或不被认可的分界符的$left，它均匀地收敛于$f$。 (3) 如果$/mathcal{U} \表示所有简单函数$u(x)$的集合，使得$f(x)\leq u(x)$对于所有$x$，并且如果$mathcal{L} \表示所有简单函数$v(x)$的集合，使得$v(x)\leq f(x)$适用于所有$x\$，那么

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