Associativity of convolution

Remarks

Both sides are formally the double integral $\iint f(y) g(z - y) h(x - z) \, dy \, dz$. Tonelli's theorem shows that $\iiint \abs{f(y) g(z - y) h(x - z)} \, dy \, dz \, dx = \norm{f}_1 \norm{g}_1 \norm{h}_1 < \infty$, so the double integral converges absolutely for almost every $x$. Fubini's theorem then allows the order of integration to be swapped, and the substitution $z = y + w$ turns one side into the other.

Example: 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 $[0, 3]$.

Integrability matters. Take $f = 1$, $g = \mathbf{1}_{[0,1]} - \mathbf{1}_{[1,2]}$ and $h = \mathbf{1}_{[0,\infty)}$ on $\R$; only $g$ is integrable. Then $f * g = \int_\R g = 0$, so $(f * g) * h = 0$. But $g * h$ is the tent function equal to $x$ on $[0, 1]$, to $2 - x$ on $[1, 2]$ and to $0$ elsewhere, so $f * (g * h) = 1$.

Going deeper

For a student

Associativity makes $L^1(\R^n)$ an associative algebra under convolution, in fact a commutative Banach algebra. It lets one define convolution powers $f^{*k}$ without brackets. In probability it says that the distribution of $X + Y + Z$ for independent $X, Y, Z$ does not depend on how the sum is grouped.