  The area of graph property testing seeks to understand the relation between
the global properties of a graph and its local statistics. In the classical
model, the local statistics of a graph is defined relative to a uniform
distribution over the graph's vertex set. A graph property $\mathcal{P}$ is
said to be testable if the local statistics of a graph can allow one to
distinguish between graphs satisfying $\mathcal{P}$ and those that are far from
satisfying it.
  Goldreich recently introduced a generalization of this model in which one
endows the vertex set of the input graph with an arbitrary and unknown
distribution, and asked which of the properties that can be tested in the
classical model can also be tested in this more general setting. We completely
resolve this problem by giving a (surprisingly "clean") characterization of
these properties. To this end, we prove a removal lemma for vertex weighted
graphs which is of independent interest.
