I am taking a real analysis course and am trying to sort out the difference between a set's measure, and a set's volume (sometimes called content).
The motivating example is a practice problem: If $A_1,A_2,…$ have volume and $A=\bigcup_{i=1}^{\infty} A_i$ is bounded, need $A$ have volume?
It turns out that this is not true, as a point has volume zero while the countable union of all rational points on $[0,1]$ does not have volume.
Volume is defined as $\int_{A}1_{A}(x) dx$
For measure, I have been using it interchangeably with volume. This this ok?
The second question I have is what is the difference between a set having volume zero and a set NOT having volume?
Thanks!
Best Answer
I think your definition of volume is the Jordan content. (Warning: This is not a real measure) Then
$$ \text{Non-measurable} \implies \text{Does not have Jordan content} $$ $$ \text{Has zero Jordan content} \implies \text{Zero measure} $$