# 数学代写|数学分析代写Mathematical Analysis代考|The Lebesgue Integral

## 数学代写|数学分析代写Mathematical Analysis代考|The Lebesgue Integral

In the previous chapter we introduced the theory of the Peano-Jordan measure and a definition of integral that essentially goes back to Riemann (1854). The years following this first definition witnessed many attempts to generalise that notion.
The theory put forward by Lebesgue at the beginning of the last century (1902) is perhaps the most sophisticated one, but without doubt the most flexible and satisfying one as well. First of all, while the Peano-Jordan measure of a subset $E \subset \mathbb{R}^n$ is defined by an approximation process involving pluri-intervals, the Lebesgue measure is built using a double approximation. The first step consists in defining the measure of an arbitrary open set $A$ by approximating it from within using pluri-intervals, and the measure of a compact set $K$ by approximating it with pluri-intervals but from outside. Next, one defines the measure of an arbitrary set $E$ by approximating it with bigger open sets and smaller compact sets. The consequence is that all open sets and all compact sets in $\mathbb{R}^n$ are “measurable”, which is not true in the Peano-Jordan theory.

Moreover, the Lebesgue measure enjoys the remarkable property of being countably additive (Theorem 1, Sect. 9.4). This, too, is false for the Peano-Jordan measure.

Particularly relevant in Lebesgue’s theory are the sets of zero measure, precisely because countable additivity guarantees that the union of a sequence of zeromeasure sets $E_k$ has zero measure. Furthermore (see Sect. 9.4) any measurable set with finite measure coincides, up to zero-measure sets, with a countable union of compact sets and with a countable intersection of open sets.

Enlarging the family of measurable sets (any Peano-Jordan measurable set is Lebesgue measurable) enables us to integrate many more functions than what we can do in Riemann’s theory. The definition of integral suggested by Lebesgue, though, has the disadvantage of not providing a concrete procedure for computing integrals explicitly. On the other hand, Fubini’s theorem allows to reduce a multiple integral to integrals in one variable.

But the Lebesgue theory’s true strength is encapsulated by the theorems ensuring when limits and integrals can be interchanged, which are way more general than what the Riemann integral allows for. These results alone justify the use of Lebesgue’s theory to address the most complex and delicate issues of Mathematical Analysis.

## 数学代写|数学分析代写Mathematical Analysis代考|Pluri-Intervals. Open Sets. Compact Sets

In the sequel we shall call
$$I=\left[a_1, b_1\right] \times\left[a_2, b_2\right] \times \ldots \times\left[a_n, b_n\right]$$
a closed interval of $\mathbb{R}^n$, and a closed pluri-interval $P$ will be a finite union of closed intervals. The measure $m(I)$ of $I$ is defined as
$$m(I)=\prod_{i=1}^n\left(b_i-a_i\right),$$
so it equals the product of the lengths of the factors $\left[a_i, b_i\right], i=1,2, \ldots, n$. If $I$ is the interval in (9.1), setting
$$I_\delta=\prod_{i=1}^n\left[a_i-\delta, b_i+\delta\right], \quad \text { with } \delta>0,$$
we have
$$I \subset \stackrel{\circ}{I}\delta, \quad \lim {\delta \rightarrow 0^{+}} m\left(I_\delta\right)=m(I)$$
That is to say: given an interval $I$, for any $\varepsilon>0$ there is an interval $J$ such that
$$I \subset \stackrel{\circ}{J}, \quad m(J)0 there is an interval J^{\prime} such that$$
$$For any i=1,2, \ldots, n, fix real numbers$$
a_{i, 0}<a_{i, 1}<\ldots<a_{i, N_i}
$$# 数学分析代考 ## 数学代写|数学分析代写Mathematical Analysis代考|The Lebesgue Integral 在上一章中，我们介绍了 Peano-Jordan 测度的理论以及本质上可追溯到 Riemann (1854) 的积分定义。在第一个定义之后的几年里，出现了许多推广该概念的尝试。 勒贝格在上个世纪初（1902 年）提出的理论可能是最复杂的理论，但无疑也是最灵活和令人满意的理论。首先，虽然子集的 Peano-Jordan 度量和⊂Rn由涉及多个区间的近似过程定义，Lebesgue 测度是使用双重近似构建的。第一步包括定义任意开集的测度A通过使用复数区间从内部对其进行近似，以及紧集的度量钾通过用多个间隔但从外部对其进行近似。接下来，定义任意集合的度量和通过用更大的开集和更小的紧集来逼近它。结果是所有开集和所有紧集Rn是“可衡量的”，这在 Peano-Jordan 理论中是不正确的。 此外，勒贝格测度具有显着的可数加性特性（定理 1，第 9.4 节）。这对于 Peano-Jordan 措施也是错误的。 与勒贝格理论特别相关的是零测度集，正是因为可数可加性保证了零测度集序列的并集和k有零措施。此外（参见第 9.4 节）任何具有有限测度的可测集，直到零测度集，都与紧集的可数并集和开集的可数交集重合。 扩大可测集族（任何 Peano-Jordan 可测集都是勒贝格可测集）使我们能够整合比我们在黎曼理论中所能做的更多的功能。然而，勒贝格建议的积分定义的缺点是没有明确提供计算积分的具体过程。另一方面，富比尼定理允许将多重积分简化为一个变量中的积分。 但勒贝格理论的真正优势在于确保何时可以互换极限和积分的定理，这比黎曼积分所允许的更为普遍。仅这些结果就证明了使用勒贝格的理论来解决数学分析中最复杂和微妙的问题是合理的。 ## 数学代写|数学分析代写Mathematical Analysis代考|Pluri-Intervals. Open Sets. Compact Sets 在续集中我们将调用$$
I=\left[a_1, b_1\right] \times\left[a_2, b_2\right] \times \ldots \times\left[a_n, b_n\right]
$$的闭区间 \mathbb{R}^n ，和一个封闭的复区间 P 将是闭区间的有限并集。的措施 m(I) 的 I 定义为$$
m(I)=\prod_{i=1}^n\left(b_i-a_i\right)
$$所以它等于因子长度的乘积 \left[a_i, b_i\right], i=1,2, \ldots, n. 如果 I 是 (9.1) 中的区间，设$$
I_\delta=\prod_{i=1}^n\left[a_i-\delta, b_i+\delta\right], \quad \text { with } \delta>0,
$$我们有$$
I \subset \stackrel{I}{\delta}, \quad \lim \delta \rightarrow 0^{+} m\left(I_\delta\right)=m(I)
$$也就是说：给定一个区间 I, 对于任何 \varepsilon>0 有间隔 J 这样$$
I \subset \stackrel{\circ}{J}, \quad m(J) 0 \text { thereisaninterval } \$J^{\prime} \text { \$suchthat }
$$Forany \ i=1,2, \ldots, n \$$, fixrealnumbers

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