Are dense subsets almost nothing or almost everything?

Solution 1:

Another twist of the answer is: uniform subsets $A$ are almost nothing or almost everything. $m(A\cap(a,b))=(b-a)\,m(A)=m(A)\,m((a,b))$ implies $m(A\cap B)=m(A)\,m(B)$ for every measurable set $B$, due the the same $\sigma$-additivity and regularity of the measure. Now with $B=A$, we obtain $m(A)=m(A)^2$, meaning $m(A)=0$ or $m(A)=1$.

Solution 2:

There certainly is no measurable example: if $E$ is measurable and of positive measure, and $0<\epsilon<1$, then there is some open interval $I$ such that $m(E\cap I)\ge\epsilon m(I)$ (this is an application of regularity, see this question). Now take $\epsilon>{1\over 2}$.