arXiv:2509.24735v4 Announce Type: replace Abstract: Bayesian updating routinely conditions on events of prior probability zero, such as an exact observation of a continuous parameter or a parameter confined to a submanifold. Two equally natural parametrizations of the same event can then return different posteriors, which is the Borel--Kolmogorov paradox. We add a metric to the probabilistic model and define the posterior given a closed null set as the limit of Maximum Entropy solutions under a vanishing constraint on the distance to that set. We give a sufficient condition for existence and prove that, given this metric, the posterior is unique and invariant under measure-preserving isometries. It recovers the textbook Bayes formulas where these apply unambiguously, on positive-measure sets and in Euclidean models. Each stage of the limit has an explicit density that depends only on the distance to the conditioning set, so the posterior can be computed with standard tools for Kullback--Leibler minimization. The construction does not remove the choice of a metric but makes it an explicit modelling input. In a controlled experiment where the geometry of the measurement is known, we show that a Bayesian analysis ignoring this geometry can yield uncalibrated posteriors. On real palaeomagnetic data, where the geometry is disputed, we show how candidate metrics can be compared within the same Bayesian framework.