A remark on real coboundary cocycles in $L^{\infty}$-space

Let $T$ be an ergodic automorphism of a probability measure space $(\Omega, {\mathcal A},m)$ and let $f$ be a real-valued measurable function on $\Omega$. We deduce a necessary and sufficient condition for the existence of $L^{\infty}$-solutions of the cohomology equation $f=h\circ T-h$, by using the recent result of Alonso, Hong and Obaya.