  In [7], Kwapie\'{n} announced that every mean zero function $f\in
L_\infty[0,1]$ can be written as a coboundary $f = g\circ T -g$ for some $g\in
L_\infty[0,1]$ and some measure preserving transformation $T$ of $[0,1]$.
Whereas the original proof in [7] holds for continuous functions, there is a
serious gap in the proof for functions with discontinuities. In this article we
fill in this gap and establish Kwapie\'{n}'s result in full generality. Our
method also allows to improve the original result by showing that for any given
$\epsilon>0$ the function $g$ can be chosen to satisfy a bound
$\|g\|_\infty\leq (1+\epsilon)\|f\|_\infty$.
