Indicator Property of the Riemann Integral

[jamilmalik]

Homework Statement

Hello fellow Mathematics enthusiasts. I was hoping someone could help me with the following problem from Terry Tao's Introduction to Measure Theory:

Let $[a,b]$ be an interval, and let $f,g:[a,b] \to \mathbb{R}$ be Riemann integrable. Establish the following statement.

(Indicator) If $E$ is a Jordan measurable of $[a,b]$, then the indicator function $1_E: [a,b] \to \mathbb{R}$ (defined by setting $1_E(x) :=1$ when $x \in E$ and $1_E(x) :=0$ otherwise.) is Riemann integrable, and $\int_{a}^{b}1_E(x) dx = m(E)$.

Homework Equations

In this problem, the notion of Jordan measure is being used. As a quick refresher, the Jordan inner measure $m_{*,(J)}(E) := \sup_{A \subset E, A \quad \text{elementary}} m(A)$ and
the Jordan outer measure $m^{*,(J)}(E) := \inf_{B \supset E, B \quad \text{elementary}} m(B)$.
Whenever $m_{*,(J)}=m^{*,(J)}$, then we say that $E$ is Jordan measurable and call $m(E)$ the Jordan measure of $E$.

The Attempt at a Solution

This is the third part of an exercise that asks the reader to establish some basic properties of the Riemann integral: linearity and monotonicity. I have done the previous two, but do not know how to start this one. Any help will be greatly appreciated, thanks in advance.

In case anyone is interested in reading the book I am using, here is the link for the free online version.

http://terrytao.files.wordpress.com/2011/01/measure-book1.pdf

 

[mighty2000]

Hello,

First, let's recall the definition of Riemann integrability. A function $f$ is said to be Riemann integrable on $[a,b]$ if for any given $\epsilon>0$, there exists a partition $P$ of $[a,b]$ such that the upper Riemann sum of $f$ over $P$ minus the lower Riemann sum of $f$ over $P$ is less than $\epsilon$.

Now, let's consider the indicator function $1_E$. Since $E$ is Jordan measurable, we know that its Jordan inner and outer measures are equal, denoted by $m_{*,(J)}(E)$ and $m^{*,(J)}(E)$ respectively. This means that there exists a sequence of elementary sets $\{A_n\}$ such that $m(A_n) \to m(E)$ as $n \to \infty$.

Now, let's choose a partition $P=\{x_0,x_1,\ldots,x_n\}$ of $[a,b]$ such that each subinterval $[x_{i-1},x_i]$ contains exactly one point from each of the sets $A_n$. This is possible since for any given subinterval, we can always choose a point from one of the sets $A_n$ that is contained in that subinterval.

Then, the upper Riemann sum of $1_E$ over $P$ is given by $U(f,P) = \sum_{i=1}^{n} \sup_{[x_{i-1},x_i]} 1_E(x_i) (x_i-x_{i-1}) = \sum_{i=1}^{n} m(E \cap [x_{i-1},x_i])$, since $1_E(x) = 1$ for all $x \in E$ and $1_E(x) = 0$ for all $x \notin E$. Similarly, the lower Riemann sum of $1_E$ over $P$ is given by ##L(f,P) = \sum_{i=1}^{n} \inf_{[x_{i-1},x_i]} 1_E(x_i) (x_i-x_{i-1}) = \sum_{i=1