I am struggling to provide sound mathematical proofs for these. Intuitively it's pretty straight forward.
Prove that the following sets is measurable and has zero area.
1) A set of a finite number of points in a plane.
Let be our finite set, and choose any
Then is a cover of the
set with length by definition, its measure is zero.
Try now to mimic the above for a finite union of finite sets in the plane, this time taking (open, say) rectangles
instead of open intervals...
2) The union of a finite collection of line segments in a plane?
Each finite line segment has width zero, so area zero. A point has length zero and height zero so it's zero.
Therefore we can say that for each line k is zero, and h is the length of the line, therefore the area is zero.
For each point h and k are zero, and therefore the area is zero.
Why are the set's measurable though?
How do we know mathematically there is one number c (zero) such that
or is this trivially demonstrated by the fact that h or k are zero above?
I want to be sure I understand this concept properly.