Problem 1495
Assume the continuum hypothesis and let \(\displaystyle \prec\) be a well-ordering of \(\displaystyle [0,1]\) of type \(\displaystyle \omega_1\). Let \(\displaystyle A = \{ (x,y)\in[0,1]^2: x\prec y \}. \) (a) Show that the horizontal sections of \(\displaystyle A\) are null-sets. (b) Show that the vertical sections of \(\displaystyle A\) have full measure. (c) Show that \(\displaystyle A\) is non-measurable with respect to 2-dimensional Lebesgue-measure. Difficulty: 6. |
Supported by the Higher Education Restructuring Fund allocated to ELTE by the Hungarian Government |