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

[Klungo]

Homework Statement

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

Homework 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)

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.

 

[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.