Fubini foiled: Katok's paradoxical example in measure theory (Q1361171)

By means of Borel's strong law of large numbers it is shown that there exist a Borel subset \(E\) of \((0,1)\times [0,1]\) with Lebesgue measure equal to one together with disjoint real analytic curves \(\Gamma_\beta\), \(\beta\in[0,1]\), which fill out \((0,1)\times [0,1]\) such that each curve \(\Gamma_\beta\) intersects \(E\) in at most a single point.