Prove that each of the following sets is measurable, and has zero area: (a) a set consisting of a single point (b) a set consisting of a finite number of points in a plane (c) the union of a finite collection of line segments in a plane(adsbygoogle = window.adsbygoogle || []).push({});

(a) To prove that a set is measurable you have to say: Let Q be a region that can be enclosed between two step regions S and T. S is a subset of Q which is a subset of T. If there is only one number c that satisfies [tex] a(S) \leq c \leq a(T) [/tex] then Q is measurable, and a(Q) = c. So a set cant be a subset of a point, so S = Q. But Q can be a subset of T. But T has to equal Q, so c = 0 and Q is measurable.

Would you do this for parts b and c?

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Measurable Sets/ Proofs: Apostol

**Physics Forums | Science Articles, Homework Help, Discussion**