}(\lceil nx \rceil , \lceil ny \rceil
)\hbox{ is an edge of }G\cr 0 & \hbox{otherwise,} \end{cases}$$ where $\lceil y \rceil$ denotes the smallest integer greater than or equal to $y$. For $g\in {{\mathcal W}}$ and simple graph $H$ we define $$t(H,g)\equiv \int_{[0,1]^\ell} \prod_{(i,j)\in E(H)}g(x_i,x_j)\,dx_1\cdots
   dx_\ell,$$ where $\ell = |V(H)|$, and note that for a graph $G$, $t(H,g^G)$ is the density of graph homomorphisms $H\to G$: $$\frac{|\hbox {hom}(H,G)|}{|V(G)|^{|V(H)|}}.$$ We define an equivalence relation on ${{\mathcal W}}$ as follows: $f\sim g$ if and only if $t(H,f)=t(H,g)$ for every simple graph $H$. Elements of ${{\mathcal W}}$ are called “graphons”, elements of the quotient space $\tilde {{\mathcal W}}$ are called “reduced graphons”, and the class containing $g\in {{\mathcal W}}$ is denoted $\tilde g$. Equivalent functions in ${{\mathcal W}}$ differ by a change of variables in the following sense. Let $\Sigma$ be the space of measure preserving maps $\sigma: [0,1]\to [0,1]$, and for $f$ in ${{\mathcal W}}$ and $\sigma\in
\Sigma$, let $f_\sigma(x,y)\equiv f(\sigma(x),\sigma(y))$. Then $f\sim
g$ if and only if there exist $\sigma, \sigma'$ in $\Sigma$ such that $f_\sigma =g_{\sigma'}$ almost everywhere; see Cor. 2.2 in [@BCL]. The space ${{\mathcal W}}$ is compact with respect to the ‘cut metric’ defined as follows. First, on 