Consistency of Bayesian estimation of a step function

We study in this paper the consistency of Bayesian estimation of a piecewise constant function defined on [0,1] with observations on a grid, using the method in Barron et al. [1999. The consistency of posterior distributions in nonparametric problems. Ann. Statist. 27, 536-561.] In particular, we show that consistency is achieved if the prior on the number of jumps is a Poisson distribution and the prior puts positive mass on the neighborhood of the true function.