# Volume of zero; Real Analysis

1. Apr 26, 2014

### alex.

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

Let $A\subset E^n$ and let $f:A\to E^m.$ Consider the condition that there exist some $M\in\mathbb{R}$ such that $d(f(x),f(y))\le Md(x,y)$ for all $x,y\in A.$

Show that if the condition is satisfied, if $m=n$, and $\text{vol}(A)=0$, then $\text{vol}((f(A))=0.$ Now suppose $m>n$ and $A$ is bounded then show that $\text{vol}(f(A))=0.$

2. Relevant equations

For an arbitrary subset $A\subset E^n,$ we say that $A$ has volume, and define the volume of $A$ to be $\text{vol}(A)=\int_A 1,$ if this integral exists.

3. The attempt at a solution

I am not sure how to do the second part of the question and I am not sure that my outline for the first part of the proof is correct.

For the first part, since the subset $A$ has volume zero then given any $\epsilon>0$ there exists a finite number of closed intervals in $E^n$ whose union contains $A$ and the sum of whose volumes is less that $\epsilon.$ So if a define $\text{vol}(A)=\int_I f$ for $A\subset I$ then for any $\epsilon>0$ there is a partition of $I$ such that any Riemann sum for $f$ corresponding to this partition has absolute value less that $\epsilon.$ So for $x,y\in I,$ let $\delta=\frac{\epsilon}{M}$ then $|f(x)-f(y)|<\epsilon.$ Then I can create step functions such that $f$ is sandwiched between the two step functions and since $\text{vol}(A)=0$ and $m=n$ then $\text{vol}(f(A))=0.$

2. Apr 26, 2014

### gopher_p

Some questions regarding the problem statment:

1) Is $E$ supposed to be some subset of $\mathbb{R}$?

2) Is the integral that you're using to define volume the Riemann integral on $\mathbb{R}^n$?

And some remarks on what you have done and questions for you to consider.

What is a closed interval in $E^n$? Are you sure this statement is even true in the case where $E=\mathbb{R}$ and $n=1$ (i.e. the simplest case)?

$\text{vol}(A)$ is already defined per your relevant equations as $\text{vol}(A)=\int_A 1$. Furthermore, considering that $f$ is a map with codomain, $E^m$, what does $\int_I f$ even mean? What does a Riemann sum for $f$ look like?

What does a step function $h:A\rightarrow E^m$ look like?

3. Apr 26, 2014

### alex.

1) Yes, it is a subset of $\mathbb{R}.$

2) That's correct

Hmm, maybe I am not sure what you mean? The phrase I used was the definition the book gave me for volume zero.

Hmm, I should start by saying what exactly is $\text{vol}(A)=\int_A 1,$ the book didn't clarify? But to answer your question, I defined $\int_I f$ to be the function $f:I\to\mathbb{R}$ by setting $f(x)=1$ if $x\in A, \ f(x)=0$ if $x\in I-A,$ so that $\text{vol}(A)=\int_I f$.

Well, I was going to define my step functions later since I didn't know if my outline was correct or not