Is the technical answer you require on page 4 of the paper you reference?

> A measure $m$ on $S_L$ is *invariant* when it is invariant under the action of $\mathfrak G_{\mathbb N}$ (the permutation group of $\mathbb N$), i.e., for every Borel set $X \subseteq S_L$ and every $g \in \mathfrak G_{\mathbb N}$, we have $m(X) = m(g.X)$.

Which means that the measure assigns the same number to isomorphic graphs.

You guess a candidate graph invariant. If you can find a simple way to do this, you will have solved [one of the great unsolved problems in computer science][1]. 


  [1]: http://en.wikipedia.org/wiki/Graph_isomorphism_problem