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# 数学代写|勒贝格积分代写Lebesgue Integration代考|MAT00013 Basic Properties of an Integral

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## 数学代写|勒贝格积分代写Lebesgue Integration代考|Basic Properties of an Integral

We will consider the value of the integral of functions in various collections. These collections all have a common domain which, for our purposes, is a closed interval. They are also closed under the operations of addition and scalar multiplication. Such a collection is a vector space of real-valued functions (see, for example, Definition A.9.1). More formally, recall that a non-empty set of real-valued functions $\mathcal{V}$ defined on a fixed closed interval is a vector space of functions provided:
(1) If $f, g \in \mathcal{V}$, then $f+g \in \mathcal{V}$.
(2) If $f \in \mathcal{V}$ and $c \in \mathbb{R}$, then $c f \in \mathcal{V}$.
Notice that this implies that the constant function 0 is in $\mathcal{V}$. All of the vector spaces we consider will contain all of the constant functions.
Three simple examples of vector spaces of functions defined on some closed interval $I$ are the constant functions, the polynomial functions, and the continuous functions.

An “integral” defined on a vector space of functions $\mathcal{V}$ is a way to assign a real number to each function in $\mathcal{V}$ and each subinterval of $I$. For the function $f \in \mathcal{V}$ and the subinterval $[a, b]$ we denote this value by $\int_a^b f(x) d x$ and call it “the integral of $f$ from $a$ to $b$.”

All the integrals we consider will satisfy five basic properties which we now enumerate.

I. Linearity: For any functions $f, g \in \mathcal{V}$, any $a, b \in I$, and any real numbers $c_1, c_2$,
$$\int_a^b c_1 f(x)+c_2 g(x) d x=c_1 \int_a^b f(x) d x+c_2 \int_a^b g(x) d x .$$
In particular, this implies that $\int_a^b 0 d x=0$.

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

The easiest functions to integrate are step functions, which we now define.

Definition 1.3.1. (Step function). A function $f:[a, b] \rightarrow \mathbb{R}$ is called a step function provided there are numbers
$$x_0=a<x_1<x_2<\cdots<x_{n-1}<x_n=b$$
such that $f(x)$ is constant on each of the open intervals $\left(x_{i-1}, x_i\right)$.
It is not difficult to see that the collection of all step functions defined on $[a, b]$ is a vector space of real-valued functions (see part (1) of Exercise 1.3.4).

We will say that the points $x_0=a<x_1<\cdots<x_{n-1}<x_n=b$ define an interval partition for the step function $f$. Note that the definition states that on the open intervals $\left(x_{i-1}, x_i\right)$ of the partition $f$ has a constant value, say $c_i$, but it says nothing about the values at the endpoints. The value of $f$ at the points $x_{i-1}$ and $x_i$ may or may not be equal to $c_i$. Of course when we define the integral this won’t matter because the endpoints form a finite set.

Since the area under the graph of a positive step function is a finite union of rectangles, it is fairly obvious what the integral should be. The $i^{t h}$ of these rectangles has width $\left(x_i-x_{i-1}\right)$ and height $c_i$ so we should sum up the areas $c_i\left(x_i-x_{i-1}\right)$. If some of the $c_i$ are negative then the corresponding $c_i\left(x_i-x_{i-1}\right)$ are also negative, but that is appropriate since the area between the graph and the $x$-axis is below the $x$-axis on the interval $\left(x_{i-1}, x_i\right)$.

## 数学代写|勒贝格积分代写Lebesgue Integration代考|积分的基本属性

1 如果$f，g在$mathcal{V}$，那么$f+g在$mathcal{V}$。

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