Prove measurability and establish that the area is zero for each of the following.
- Any set consisting of a single point.
- Any set consisting of finitely many points in the plane.
- The finite union of line segments in the plane.
- Proof. Since all rectangles are measurable with area equal to where and are the lengths of the edges of the rectangle, a single point is measurable with area 0, since a point is rectangle with
Proof. We prove by induction on , the number of points. For the case , the statement is true by part (a). Now, assume it is true for some . Then, we have a set of points in the plane and . Let be a point in the plane. By part (a), and . Thus,
But, , so
(Where Axiom 1 of area guarantees us that cannot be negative.) Thus, . Hence, the statement is true for points in a plane, and thus for all
Proof. We again use induction, now on , the number of lines in a plane. For , we let be a set with one line in a plane. Since a line is rectangle and all rectangles are measurable, we have . Further, since a line is a rectangle with either or , and so in either case, . Thus, the statement is true for a single line in the plane, the case .
Assume then that it is true for . Let be a set of lines in the plane. Then by the induction hypothesis, and . Let be a single line in the plane. By the case above, and . Thus, and (since ). Hence, the statement is true for lines in a plane, and so for all