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

Equidistribution in measure-preserving actions of semisimple groups : case of $SL_2(\mathbb{R})$