  We introduce probability-graphons which are probability kernels that
generalize graphons to the case of weighted graphs. Probability-graphons appear
as the limit objects to study sequences of large weighted graphs whose
distribution of subgraph sampling converge. The edge-weights are taken from a
general Polish space, which also covers the case of decorated graphs. Here,
graphs can be either directed or undirected. Starting from a distance $d_m$
inducing the weak topology on measures, we define a cut distance on
probability-graphons, making it a Polish space, and study the properties of
this cut distance. In particular, we exhibit a tightness criterion for
probability-graphons related to relative compactness in the cut distance. We
also prove that under some conditions on the distance $d_m$, which are
satisfied for some well-know distances like the Prohorov distance, and the
Fortet-Mourier and Kantorovitch-Rubinstein norms, the topology induced by the
cut distance on the spaceof probability-graphons is independent from the choice
of $d_m$. Eventually, we prove that this topology coincides with the topology
induced by the convergence in distribution of the sampled subgraphs.
