3
$\begingroup$

Let $f$ be Lebesgue measurable in $[0,1]$ and assume $f$ takes finitely many values. Assuming $f(x) - f(y)$ is Lebesgue measurable in $[0,1] \times [0,1]$ show that $f$ is integrable over $[0,1]$.

Stuck for a while with this one. (Not homework, just practice)

1 Answers 1

4

The fact that $f(x)-f(y)$ is Lebesgue measurable implies that it is defined almost everywhere on the square. For this to be possible, $f$ cannot be $+\infty$ or $-\infty$ on a set $S$ of measure greater than 0 (otherwise $f(x)-f(y)$ will be undefined on the set $S\times S$ of positive measure). Since $f$ also takes on only finitely values, it is integrable.