# Prove a set consisting of a single point is measurable and has zero area

1. Feb 3, 2012

### Klungo

1. The problem statement, all variables and given/known data
Prove that a set consisting of a single point is measurable and has zero area.

2. Relevant equations
Area Axioms: $$M$$ is a class of measurable sets.
(a) Every rectangle $$R \in M$$. If the edges of R have lengths h and k, then the area $$a(R) = hk$$.

Additionally, a rectangle can be represented as $$R=((x,y)|0≤x≤h,0≤y≤k)$$

(b) If a set $$S \in M$$ and if $$S$$ is congruent to $$T$$, then $$T \in M$$ and the areas $$a(S) = a(T)$$

3. The attempt at a solution

Let $$(x_{0},y_{0}) \in ((x_{0},y_{0}))$$ be an arbitrary point on the plane. Since $$((x_{0},y_{0})) = ((x,y)|x_{0}≤x≤x_{0},y_{0}≤y≤y_{0})$$ (a) which is a rectangle $$R$$ and so $$((x_{0},y_{0})) \in M$$ (measurable) by (b). Since the side lenghts of $$((x_{0},y_{0}))$$ are zero, then $$a(((x_{0},y_{0}))) = 0$$ .

Is this right? I apologize for my tex.

2. Feb 3, 2012

### HallsofIvy

Yes, that is correct. You might want to specify that the side lengths of the "rectangle" are $x_0- x_0$ and $y_0- y_0$ which are, as you say, 0.