  Bayesian inverse problem on an infinite dimensional separable Hilbert space
with the whole state observed is well posed when the prior state distribution
is a Gaussian probability measure and the data error covariance is a cylindric
Gaussian measure whose covariance has positive lower bound. If the state
distribution and the data distribution are equivalent Gaussian probability
measures, then the Bayesian posterior measure is not well defined. If the state
covariance and the data error covariance commute, then the Bayesian posterior
measure is well defined for all data vectors if and only if the data error
covariance has positive lower bound, and the set of data vectors for which the
Bayesian posterior measure is not well defined is dense if the data error
covariance does not have positive lower bound.
