A set $S \subset \mathbb{R}$ is considered to have zero content if for any $\epsilon>0$ there is a finite collection of intervals $I_1,…,I_L$ such that
i) $S\subset \bigcup_{i=1}^L I_i$ and
ii) the sum of the lengths of the
$I_i$'s is less than $\epsilon$.
I am currently an undergraduate student taking an introductory course and have not covered Lebesgue measure. Can anyone provide examples of sets that do not have zero content and sets that do have zero content?
Best Answer
I guess the interesting question is to find sets of measure $0$ but content non-zero (since everythng that has content $0$, definitely also has measure $0$...)
The simplest, I think, example is any unbounded set of $0$ measure. It cannot have $0$ content, because it has to be covered by finitely many intervals, so one of them has to have infinite length...
More examples: In the bounded case, consider the rational numbers inside $[0,1]$. This set definitely has measure $0$ but content $1$. On the other hand, a compact set of measure $0$ will have content $0$ as well (because any open cover has a finite subcover, etc...).