Many scientific and engineering problems require to perform Bayesian inferences in
function spaces, in which the unknowns are of infinite dimension. In such problems,
choosing an appropriate prior distribution is an important task. In particular we
consider problems where the function to infer is subject to sharp jumps which render
the commonly used Gaussian measures unsuitable. On the other hand, the so-called total
variation (TV) prior can only be defined in a finite dimensional setting, and does
not lead to a well-defined posterior measure in function spaces. In this work we present
a TV-Gaussian (TG) prior to address such problems, where the TV term is used to detect
sharp jumps of the function, and the Gaussian distribution is used as a reference
measure so that it results in a well-defined posterior measure in the function space.
We also present an efficient Markov Chain Monte Carlo (MCMC) algorithm to draw samples
from the posterior distribution of the TG prior. With numerical examples we demonstrate
the performance of the TG prior and the efficiency of the proposed MCMC algorithm.