I agree with @Gjergji that the technical answer you require is 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 $m(X) = m(g.X)$.

So, to your friend, you can say that "invariant measure" means that the measure assigns the same number to isomorphic graphs.

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

I agree with @Gjergji that the technical answer you require is 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 $m(X) = m(g.X)$.

So, they put a measure m on the space of your special graphs. For any set of graphs $X \subseteq S_L$, we measure the size of this set as $m(X)$. If we rename all the verticies of $X$ and call this set $g.X$, then these sets have equal measure $m(X)=m(g.X)$.

So, to your friend, you can say that "invariant measure" means that the measure assigns the same number to sets of isomorphic graphs.

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.

I agree with @Gjergji that the technical answer you require is 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 $m(X) = m(g.X)$.

So, to your friend, you can say that "invariant measure" means that the measure assigns the same number to isomorphic graphs.

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