  The paper formulates Bayesian inverse problems for inference in a topological
measure space given noisy observations. Conditions for the validity of the
Bayes formula and the well-posedness of the posterior measure are studied. The
abstract theory is then applied to Burgers and Hamilton-Jacobi equations on a
semi-infinite time interval with forcing functions which are white noise in
time. Inference is made on the white noise forcing, assuming the Wiener measure
as the prior.
