Coboundaries of commuting Borel automorphisms

We show that if $S,T$ are two commuting automorphisms of standard Borel space such that they generate a free Borel ${\mathbb{Z}}^2$-action then $S$ and $T$ do not have same sets of real valued bounded coboundaries. We also prove a weaker form of Rokhlin Lemma for Borel ${\mathbb{Z}}^d$-actions.