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.