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?

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# Measurable Sets/ Proofs: Apostol

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