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  "body": "\n## Statement\n\nLet $n \\ge 1$, and let $f, g, h \\in L^1(\\R^n)$.\nThen the [[convolutions|2Z29JN3VES]] $(f * g) * h$ and $f * (g * h)$ are both defined almost everywhere, they lie in [[$L^1(\\R^n)$|SASP14MA1Z]], and\n$$(f * g) * h = f * (g * h)$$\nalmost everywhere.\n\n## Remarks\n\nBoth sides are formally the double integral $\\iint f(y) g(z - y) h(x - z) \\, dy \\, dz$.\n[[Tonelli's theorem|067SYD2J4B]] shows that $\\iiint \\abs{f(y) g(z - y) h(x - z)} \\, dy \\, dz \\, dx = \\norm{f}_1 \\norm{g}_1 \\norm{h}_1 \u003c \\infty$, so the double integral converges absolutely for almost every $x$.\n[[Fubini's theorem|0R0T1VWH5X]] then allows the order of integration to be swapped, and the substitution $z = y + w$ turns one side into the other.\n\nExample: on $\\R$ with $f = g = h = \\mathbf{1}_{[0, 1]}$, both sides equal the convolution of the tent function $\\mathbf{1}_{[0,1]} * \\mathbf{1}_{[0,1]}$ with $\\mathbf{1}_{[0, 1]}$, a piecewise quadratic spline supported on $[0, 3]$.\n\nIntegrability matters.\nTake $f = 1$, $g = \\mathbf{1}_{[0,1]} - \\mathbf{1}_{[1,2]}$ and $h = \\mathbf{1}_{[0,\\infty)}$ on $\\R$; only $g$ is integrable.\nThen $f * g = \\int_\\R g = 0$, so $(f * g) * h = 0$.\nBut $g * h$ is the tent function equal to $x$ on $[0, 1]$, to $2 - x$ on $[1, 2]$ and to $0$ elsewhere, so $f * (g * h) = 1$.\n\n## Explanations\n\n### L1 human\n\nWhen you convolve three functions, you may first combine the first two or the last two.\nThe result is the same.\nSo an expression like $f * g * h$ needs no brackets.\n\n### L2 human\n\nAssociativity makes $L^1(\\R^n)$ an associative algebra under convolution, in fact a commutative Banach algebra.\nIt lets one define [[convolution powers|JGR1E7Q9KQ]] $f^{*k}$ without brackets.\nIn probability it says that the distribution of $X + Y + Z$ for independent $X, Y, Z$ does not depend on how the sum is grouped.\n",
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