Ergodic averages and free ${\displaystyle {\mathbb{Z}}^2}$ actions

If the ergodic transformations S, T generate a free ${\displaystyle {\mathbb{Z}}^2}$ action on a finite non-atomic measure space (X,S,µ) then for any ${\displaystyle c_1,c_2\in ℝ}$ there exists a measurable function f on X for which ${\displaystyle {\left(\{N+1\}\right)}^{-1}{\sum }_{j=0}^{N}f\left(S^{j}x\right)\to c_1}$ and ${\displaystyle {(N+1)}^{-1}{\sum }_{j=0}^{N}f\left(T^{j}x\right)\to c_2\mu }$-almost everywhere as N → ∞. In the special case when S, T are rationally independent rotations of the circle this result answers a question of M. Laczkovich.