# A proof of an area as a set function

1. Nov 16, 2009

### Shing

Hi guys, recently, I am a freshmen majoring in Physics, recently using Apostle Calculus to self-study. However, I am having a hard time knowing whether if my proof is correct or not, since there isn't any solution.
1. The problem statement, all variables and given/known data
Prove that following set is measurable and has zero area: a set consisting of a single point.

3. The attempt at a solution

First, I have considered the Axiom of Choice of scale{Every rectangle R is in Measurable Set. If the edges of R have lengths h and k then a(R)=hk} to show that h and k both is zero therefore, a(Point)=0

but I couldn't convince myself that a point is an rectangle!! So my former approach is not proper.

and I couldn't think up any other approaches so far.

would anyone be king enough to give me a few suggestions?

many thanks!!

2. Nov 16, 2009

### Office_Shredder

Staff Emeritus
Think about making really small rectangles around the point

3. Nov 29, 2009

### Shing

Thank you very much indeed!!

so here is my new approach, what would the problem be in my proof? :
let there be a point in a rectangle $a^2$
$$\exists a\in R$$
s.t. for all $x\in R, 0<a<x$
$$\implies a^2=0$$
$$\implies f(point)=0$$