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  "body": "\n## Definition\n\nA *graphon* is a [[measurable function|YYNCJSR0PJ]] $W \\colon [0, 1]^2 \\to [0, 1]$, for the [[Lebesgue measure|2Z2QVWZ262]] on $[0, 1]^2$, such that $W(x, y) = W(y, x)$ for all $x, y \\in [0, 1]$.\n\n## Remarks\n\nExample: let $W(x, y) = 1$ if exactly one of $x, y$ is less than $1/2$, and $W(x, y) = 0$ otherwise.\nIt takes values in $\\set{0, 1}$, and it is measurable because it is constant on four rectangles.\nIt is symmetric, since swapping $x$ and $y$ does not change whether exactly one of them is below $1/2$.\nIt is the limit of the complete bipartite graphs $K_{n,n}$.\n\nNon-example: $W(x, y) = x$ is measurable with values in $[0, 1]$, but it is not symmetric, since $W(1, 0) = 1 \\ne 0 = W(0, 1)$.\n\nA graph $G$ on $\\set{1, \\ldots, n}$ gives a step graphon $W_G$: $W_G(x, y) = 1$ if $\\ceil{nx}$ and $\\ceil{ny}$ are adjacent in $G$, and $0$ otherwise (with $\\ceil{0}$ read as $1$).\nFor a finite graph $F$ on $\\set{1, \\ldots, k}$, the homomorphism density of $F$ in $W$ is\n$$t(F, W) = \\int_{[0,1]^k} \\prod_{\\set{i, j} \\in E(F)} W(x_i, x_j) \\, dx_1 \\cdots dx_k.$$\nGraphons that differ on a set of measure zero, or by a measure-preserving change of variables, are usually identified.\nSome authors allow any standard probability space in place of $[0, 1]$.\n",
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