  Guided by the theory of graph limits, we investigate a variant of the cut
metric for limit objects of sequences of discrete probability distributions.
Apart from establishing basic results, we introduce a natural operation called
{\em pinning} on the space of limit objects and show how this operation yields
a canonical cut metric approximation to a given probability distribution akin
to the weak regularity lemma for graphons. We also establish the cut metric
continuity of basic operations such as taking product measures.
