Weak and TV consistency in Bayesian uncertainty quantification using disintegration

Using standard techniques in Probability theory we prove a series of results relevant in the theory of Bayesian uncertainty quantification (UQ). Using the approach, found in the Bayesian literature, of defining the posterior distribution through a disintegration argument, and using weak and total variation convergence, we are able to prove the existence and numerical consistency of the posterior measure in general functional (Banach) spaces. Relaying commonly on simpler proofs and weaker assumptions, we establish these basic results useful for the theoretical foundation of most common and current UQ problems.