Consider the Borel-sigma algebra $([0,1]^2, \mathcal{B}([0,1]^2))$. Now I struggle to find a measure (other than the Lebesgue-Borel measure) that fulfills $\mu([a,b]\times[0,1])=b-a$ and $\mu([0,1]\times[a,b])=b-a$.
Whenever I construct a measure, I violate the property of additivity for disjoint sets when considering the union of sets. E.g. I tried to define the measure such that it gives the length of the shorter side of a rectangle. However, this clearly does not work for all unions of elements in $\mathcal{B}([0,1]^2)$.
I would be extremely thankful for any hint.