Indicator Property of the Riemann Integral

1. Sep 7, 2014

jamilmalik

1. The problem statement, all variables and given/known data

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 $\int_{a}^{b}1_E(x) dx = m(E)$.

2. Relevant 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 $E$.

3. 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

Last edited: Sep 7, 2014