# Homework Help: Measuring Unsigned Simple Functions

1. Sep 22, 2014

### jamilmalik

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

I was hoping that someone would be able to help me solve this problem regarding simple functions and their measure. This problem is coming straight from *Introduction to Measure Theory* by Terrence Tao. A link to the free version is attached at the end of this post.

Show that an unsigned function $f: \mathbb{R}^d \to [0, +\infty]$ is a simple function if and only if it is measurable and takes on at most finitely many values.

2. Relevant equations

The definitions that I am working with are as follows:

An *unsigned simple function* $f: \mathbb{R}^d \to [0, +\infty]$ is a finite linear combination $f= c_11_{E_1}+ \ldots + c_k1_{E_k}$ of indicator functions $1_{E_i}$ of Lebesgue measurable sets $E_i \subset \mathbb{R}^d$ for $i=1, \ldots , k,$ where $k \geq 0$.

An unsigned simple function $f:\mathbb{R}^d \to [0, +\infty]$ is *unsigned Lebesgue measurable*, or *measurable* for short, if it is the point-wise limit of unsigned simple functions, i.e., if there exists a sequence $f_1,f_2,f_3, \ldots : \mathbb{R}^d \to [0, +\infty]$ of unsigned simple functions such that $f_n(x) \to f(x)$ for every $x \in \mathbb{R}^d$.

3. The attempt at a solution

My attempt at the solution is this:

If $f:\mathbb{R}^d \to [0, +\infty]$ is simple, then $f$ is a linear combination $f=c_11_{E_1}+ \ldots + c_n1_{E_n}$ of indicator functions $1_{E_i}$ of Lebesgue measurable sets $E_i \subset \mathbb{R}^d$ for $i =1, \ldots, n$, where $n \in \mathbb{N}$ and $c_1, \ldots, c_n \in [0, +\infty]$. We wish to show that this implies that $f$ is measurable and takes on at most finitely many values. To say that it is measurable would mean that $\exists f_1, f_2, \ldots$ each unsigned simple such that $\forall x \in \mathbb{R}^d, f(x) = \displaystyle \lim_{n \to \infty} f_n(x)$. If we cut up $[0, +\infty]$ into finitely many boxes $B$, each small enough that $\forall x, y \in B, |f(x)-f(y)| < \frac{1}{n}$, or $f(x), f(y) \in [0, +\infty].$ $f_n(x) = \inf(f(B))$ if $x \in$ box $B$. $f_n(x)=0$ for $x \notin [0, +\infty]$.

Then $\forall x \in \mathbb{R}^d$,
$f_n(x) \to f(x)$ and $f_n = \displaystyle \sum_{\text{boxes B}}\inf(f(B))1_B$ is simple.

To prove the backwards direction, if $f$ is measurable, then $\exists f_1,f_2 \ldots$ each unsigned simple such that $\forall x\in \mathbb{R}^d, f(x)=\displaystyle \lim_{n=\infty}f_n(x)$. Since $f$ is composed of simple functions converging point-wise, the result follows by definition.

Note: In this proof, I am imitating an approach we used in class to show that if a function is continuous, then it is measurable. I am not sure if it still applies in this case, which is why I would really appreciate some feedback in order to help me solve this problem.

Thank you very much in advance.