I'd like to understand the geometric abstraction of measure in more depth, and my contemplations about it led me to a number of conclusions:
- A geometric point is infinitely small in relation to any geometric space, as there is no way to measure a point.
- Any line passes through infinitely many points.
- A line is infinitely big in relation to one-dimensional space, but also infinitely small in relation to any space of higher dimension.
Clearly, the fact that the line passes through more than one point should lead to the conclusion that the line has bigger measure than the single point, but both are infinitely small in relation to $n$-dimensional space with $n\gt 1$.
How to interpret the measure of a line correctly? How to interpret the measure of $n-1$-dimensional subspace of $n$-dimensional space in general?
The measure of a set in $n$ space has little relation to the number of points it contains.
To get started on how to calculate the measure of a set, think about rectangles. The measure of a rectangle is its area: the product of its length and width.
Now imagine a line segment a mile long. Even though it has infinitely many points, it will have measure $0$. To see why, enclose that segment in a rectangle with length $1$ mile and width $0.001$ miles. The area of that rectangle is $0.001$ square miles. So the measure of that line is less than $0.001$ square miles. There's nothing special about $0.001$ in this argument, so the measure of the line is smaller than any positive number, so it must be $0$.
To make this argument work for an infinite line you have to work just a little harder. Surround it by an infinite sequence of rectangles of fixed length whose width decrease in such a way as to make the sum of the areas converge to as small a number as you wish.
In $n$ space the measure of a box is the product of the lengths of its sides. You can reason similarly to show that subspaces of lower dimension in $n$ space have measure $0.