(adsbygoogle = window.adsbygoogle || []).push({}); 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: [tex] M [/tex] is a class of measurable sets.

(a) Every rectangle [tex]R \in M [/tex]. If the edges of R have lengths h and k, then the area [tex] a(R) = hk [/tex].

Additionally, a rectangle can be represented as [tex] R=((x,y)|0≤x≤h,0≤y≤k) [/tex]

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

3. The attempt at a solution

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

Is this right? I apologize for my tex.

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

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