# 数学代写|实分析作业代写Real analysis代考|MATH2350

#### Doug I. Jones

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The term “scalar functions” will be reserved for functions with values in $(-\infty,+\infty)$. In those cases where we consider functions with values in the extended real line $[-\infty,+\infty]$, this will be particularly notified.
2.3.1. Definition. A scalar set function $\mu$ defined on some class of sets $\mathcal{A}$ is called additive if
$$\mu\left(\bigcup_{i=1}^n A_i\right)=\sum_{i=1}^n \mu\left(A_i\right)$$
for all finite collections of pairwise disjoint sets $A_i \in \mathcal{A}$ such that $\bigcup_{i=1}^n A_i \in \mathcal{A}$. The function $\mu$ is called countably additive if
$$\mu\left(\bigcup_{n=1}^{\infty} A_n\right)=\sum_{n=1}^{\infty} \mu\left(A_n\right)$$
for all countable collections of pairwise disjoint sets $A_n$ in $\mathcal{A}$ with $\bigcup_{n=1}^{\infty} A_n \in \mathcal{A}$. A countably additive set function defined on an algebra is called a measure.

A countably additive measure $\mu$ on a $\sigma$-algebra of subsets of a space $X$ is called a probability measure if $\mu$ is nonnegative and $\mu(X)=1$.

A measure defined on the Borel $\sigma$-algebra of the whole space $\mathbb{R}^n$ or of its part is called a Borel measure.

It is easily seen from the definition that the series (2.3.2) converges absolutely (since its sum does not depend on rearrangements of the series).

## 数学代写|实分析作业代写Real analysis代考|The Outer Measure and the Lebesgue Extension of Measures

Here we show how to extend countably additive measures from algebras to $\sigma$-algebras. On extensions from rings, see $\$ 2.7($i). We shall consider finite set functions and in the end make a remark about functions with values in$[0,+\infty]$. For every nonnegative set function$\mu$defined on some class$\mathcal{A}$of subsets of a space$X$containing$X$itself the formula $$\mu^(A)=\inf \left{\sum_{n=1}^{\infty} \mu\left(A_n\right) \mid A_n \in \mathcal{A}, A \subset \bigcup_{n=1}^{\infty} A_n\right}$$ defines a new set function defined for every subset$A \subset X$. The same construction applies to set functions with values in$[0,+\infty]$. If$X$does not belong to$\mathcal{A}$, the function$\mu^$is defined by the indicated formula on all sets$A$that can be covered by countable sequences of elements of$\mathcal{A}$, and to all other sets one can assign the infinite value (sometimes it is more convenient to assign them the value equal the supremum of values of$\mu^$on their subsets that can be covered by sequences from$\mathcal{A}$). The function$\mu^$is called the outer measure generated by$\mu$, although it need not be even additive. In greater detail Caratheodory outer measures, not necessarily generated by additive set functions, are discussed in [73, Chapter 1]; see also$\$2.7$ (i) below. Our main example (Lebesgue measure): $A \subset[0,1]$ is covered by intervals $A_n$ and $\mu\left(A_n\right)$ is the length of $A_n$, see $\S 2.5$.
2.4.1. Definition. Let $\mu$ be a nonnegative set function on some domain of definition $\mathcal{A} \subset 2^X$. A set $A$ is called $\mu$-measurable (or Lebesgue measurable with respect to $\mu$ ) if, for every $\varepsilon>0$, there is a set $A_{\varepsilon} \in \mathcal{A}$ with $\mu^\left(A \triangle A_{\varepsilon}\right)<\varepsilon$. The class of $\mu$-measurable sets is denoted by $\mathcal{A}_\mu$. We shall be interested in the case where $\mu$ is a countably additive measure on an algebra $\mathcal{A}$. The definition of measurability given by Lebesgue himself consisted in the equality $\mu^(A)+\mu^*(X \backslash A)=\mu(X)$ (for a closed interval $X$ ). It will be shown below that for additive functions on algebras this definition (possibly, intuitively not that transparent, but simply expressing the additivity for mutually complementing sets) is equivalent to the one given above (see Proposition 2.4.12 and Theorem 2.7.8). In addition, the cited assertions contain a criterion of the Caratheodory measurability, which is also equivalent to our definition in the case of nonnegative additive set functions on algebras, but is much more effective in the general case (in particular, for measures with values in $[0,+\infty]$ ).

# 实分析代写

2.3.1. 定义。标量集函数 $1 m u \mu$ 定义在某类集合 Imathcal ${A} \mathcal{A}$ 称为加法，如果
$$\mu\left(\bigcup_{i=1}^n A_i\right)=\sum_{i=1}^n \mu\left(A_i\right)$$

$$\mu\left(\bigcup_{n=1}^{\infty} A_n\right)=\sum_{n=1}^{\infty} \mu\left(A_n\right)$$

## 数学代写|实分析作业代写Real analysis代考|The Outer Measure and the Lebesgue Extension of Measures

2.4.1. 定义。让 $\mu$ 是某个定义域上的非负集函数 $\mathcal{A} \subset 2^X$. 一套 $A$ 叫做 $\mu$-可测量的 (或关于勒贝格可测 量的 $\mu$ ) 如果，对于每个 $\varepsilon>0$, 有一个集合 $A_{\varepsilon} \in \mathcal{A}$ 和 $\mu^{\left(A \triangle A_{\varepsilon}\right)}<\varepsilon$. \mu类 $\mu$-可测集表示为 $\mathcal{A}_\mu$. 我们会对以

$\left.\mu^{(} A\right)+\mu^*(X \backslash A)=\mu(X)$ (对于闭区间 $X$ ). 下面将 证明，对于代数上的加法函数，这个定义 (可能，直观 上不是那么透明，而是简单地表达了相互补充集的可加 性) 等同于上面给出的定义 (见命题 2.4.12 和定理
2.7.8）. 此外，引用的断言包含 Caratheodory 可测性 标准，这也等同于我们在代数上非负加性集函数的情况 下的定义，但在一般情况下更有效 (特别是对于值为 $[0,+\infty])$

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