SCHWEITZER 2002 Problem 5

Denote by $\lambda (H)$ the Lebesgue outer measure of $H\subseteq \left[ 0,1\right]$ . The horizontal and vertical sections of the set $A\subseteq [0, 1]\times [ 0, 1]$ are denoted by $A^y$ and $A_x$ respectively; that is, $A^y=\{ x\in [ 0, 1] \colon (x, y) \in A\}$ and $A_x=\{ y\in [ 0, 1]\colon (x,y)\in A\}$ for all $x,y\in [0,1]$ . (a) Is there a decomposition $A\cup B$ of the unit square $[0,1]\times [0,1]$ such that $A^y$ is the union of finitely many segments of total length less than $\frac12$ and $\lambda (B_x)\le \frac12$ for all $x, y\in [0,1]$ ? (b) Is there a decomposition $A\cup B$ of the unit square $[0,1] \times [0,1]$ such that $A^y$ is the union of finitely many segments of total length not greater than $\frac12$ and $\lambda (B_x)<\frac12$ for all $x,y\in [0,1]$ ?