Abstract: We present conditions for Bayesian consistency in the supremum metric. The key to the technique is a triangle inequality that allows us to explicitly use weak convergence, a consequence of the standard Kullback–Leibler support condition for the prior. A further condition is to ensure that smoothed versions of densities are not too far from the original density, thus dealing with densities that could track the data too closely. Our main result is that we demonstrate supremum consistency using conditions comparable with those currently used to secure 𝕃₁-consistency.

Key words and phrases: Fourier integral theorem, Prokhorov metric, sinc kernel, weak convergence.