Proving Measurability and Zero Area for Finite Sets in a Plane

[courtrigrad]

Lets say we have the following sets: (a) set consisting of a single point (b) set consisting of finite number of points in a plane (c) union of a finite collection of line segments in a plane. We want to prove that each of these sets is measurable and has zero area. Ok so here is how I started:

So for (a) Q is a step that can be enclosed between two step regions S and T so that there is one c which satisfies the inequalities $$a(S) \leq c \leq a(T)$$ for all regions S and T satisfying this then Q is measurable and $$a(q) = c$$ So should I choose c = 0? This will be both less than and greater than two given areas. Should I do the same thing for the other parts?

Thanks a lot

 

[courtrigrad]

any help would be appreciated

 

[HallsofIvy]

Looks to me like they are pretty straight forward. In fact, you don't need to think about "measure" in general. For the first one, a single point, just show that a single point can be contained in intervals of arbitrarily small length.
For the second, a finite number of points, just use "summability" (what is the sum of a finite number of 0s?) or take a small rectangle about each point- and add those. Show that the rectangles can be taken to be arbitrarily small.
For the third, line segments, surround each segment by rectangles of the same length as the segment and of smaller and smaller widths.