Level sets of depth measures in abstract spaces.

The lens depth of a point has been recently extended to general metric spaces, which is not the case for most depths. It is defined as the probability of being included in the intersection of two random balls centred at two random points and , with the same radius (, ). We prove that, on a separable and complete metric space, the level sets of the empirical lens depth based on an iid sample, converge in the Painlevé-Kuratowski sense, to its population counterpart.