MathDB

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]

Problem Statement