I don't understand. What makes a set have measure 0 or not?

>>381087
You're asking what it means for a set to have measure 0, but it's in your text, so you could just have looked there.
Anyway, the definition is:
>A is of measure zero if for all ε > 0 there exists a sequence of intervals whose union contains A and whose total length is less than ε.

Most texts give the definition of "has measure zero" before defining the (Lebesgue) measure.

So the question is: if A can't be covered by intervals of arbitrarily small total length, but B can, prove that A \ B can't.