Uniform L1 posterior consistency in compact Gaussian shift experiments

For a suitably chosen metric ρ for the topology of weak convergence in the space of prior distributions on the shift parameter of a compact Gaussian shift experiment, the posterior distribution (induced by a full support prior) of the shift parameter given independent and identically distributed observations from the experiment is uniformly (in the shift parameter) L1 consistent in ρ. The corresponding posterior mean is uniformly L1 consistent in the norm on the parameter space. ρ-distance between the posterior mean (induced by a full support hyperprior) of the prior (i.e., mixing) distribution given independent and identically distributed observations from the mixture experiment and the empirical distribution of the parameter sequence in the product experiment [the mixing distribution] goes to zero in L1 of the product [mixture] experiment, uniformly in parameter sequences [mixing distributions].