Equidistribution in measure-preserving actions of semisimple groups : case of SL₂(mathbb{R})

We prove pointwise convergence for the semi-radial averages on $G=SL_2(\mathbb{R})$ given by $\int_t^{t+1} m_K\ast \delta_{a_{s}}ds$ (and similar variants), acting on $K$-finite $L^p$-functions in a probability-measure-preserving action of the group, for $p > 1$.