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In [7], Kwapie\'{n} announced that every mean zero function $f\inL_\infty[0,1]$ can be written as a coboundary $f = g\circ T -g$ for some $g\inL_\infty[0,1]$ and some measure preserving transformation $T$ of $[0,1]$.Whereas the original proof in [7] holds for continuous functions, there is aserious gap in the proof for functions with discontinuities. In this article wefill in this gap and establish Kwapie\'{n}'s result in full generality. Ourmethod 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$\|g\|_\infty\leq (1+\epsilon)\|f\|_\infty$.

Full proof of Kwapie\'n's theorem on representing bounded mean zero functions on $[0,1]$