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We report on an indirect and non-invasive method to simultaneously characterise the energy-dependent emittance and source size of ultra-relativistic positron beams generated during the propagation of a laser-wakefield accelerated electron beam through a high-Z converter target. The strong correlation of the geometrical emittance of the positrons with that of the scattered electrons allows the former to be inferred, with high accuracy, from monitoring the latter. The technique has been tested in a proof-of-principle experiment where, for 100 MeV positrons, we infer geometrical emittances and source sizes of the order of 3 µm and 150 µm, respectively. This is consistent with the numerically predicted possibility of achieving sub-µm geometrical emittances and micron-scale source sizes at the GeV level. [ABSTRACT FROM AUTHOR]