Probability measures on distribution spaces and quantum field theoretical models

It is well known that there is a close connection between (Abelian) fields and probability measures on distribution space, or more generally, on infinite-dimensional vector spaces, and their associated random processes. We establish general criteria for mutual singularity of such measures and apply them to quantum fields. In particular, it is shown that a kind of generalized clustering implies singularity. Then a condition for singularity is given in terms of a natural metric introduced previously. It is used to show that the translate of a measure by a linear functional which is not continuous for this metric is singular to the untranslated measure. The results are applied to processes with independent values at each point, corresponding to the ultra-local model. It is shown that each such measuris singular to any of its translates although its finite-dimensional projections may be equivalent to Lebesgue measure.