There are subsets of the real line that has infinite counting measure, but Lebegue measure 0, so the Lebegue measure is used for measuring larger sets than the counting measure. My question is: Is there a translation invariant measure m such that for some sets with Lebegue measure 0 the m-measure is infinite and for some sets with infinite counting measure, the m-measure is 0?
I have found one example: m(A)=0 if A is countable, and m(A)=infinite otherwise. So I will require that the measure can take the value 1.
If such a measure exist, can we find a measure between this and the counting measure? and between this and the Lebegue measure? and so on.