Problem 1495 (difficulty: 6/10)

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.

Give me another random problem!