Suppose that \(X\) and \(Y\) are independent random variables with distribution \(\mu\) and \(\nu\), respectively, and let \(H\in \mathcal{R}\). Then by Fubini’s theorem, \[\begin{eqnarray*} P(X+Y\in H)&=& \int_{\Omega_1\times\Omega_2}I(X(\omega_1)+Y(\omega_2)\in H)d\mu\times\nu(\omega_1\times \omega_2) \\ &=&\int_{\mathbb{R}^2}I(x+y\in H)\mbox{ }d \mu\times\nu(x\times y) \mbox{ }\mbox{ }(\mbox{basically, }\mu_X\times\mu_Y)\\ &=&\int_{\mathbb{R}^2}I(x+y\in H)\mbox{ }d \mu(x)d\nu(y) \\ &=&\int_{\mathbb{R}}\left[\int_{\mathbb{R}}I(y\in H-x)\mbox{ }d\nu(y)\right]d \mu(x)\\ &=&\int_{\mathbb{R}}\nu(H-x)d\mu(x), \end{eqnarray*}\] where \(H-x=\{ z-x:z\in H\}\).
In more intuitive notation, for independent \(X\) and \(Y\), if \(P_X,P_Y\) and \(P_{X+Y}\) represent the distributions of \(X,Y\) and \(X+Y\), respectively, then \[
P_{X+Y}(H)=\int_\mathbb{R}P_Y(H-x)\mbox{ }dP_X(x)
\]
If \(\mu\) and \(\nu\) are probability measures on \((\mathbb{R}, \mathcal{R})\), then the measure \(\mu*\nu\) defined by \[ (\mu*\nu)(H)=\int_{-\infty}^\infty \nu(H-x)\mbox{ }d\mu(x), \mbox{ }\mbox{ }\mbox{ }\mbox{ }H\in \mathcal{R}, \] is called the convolution of \(\mu\) and \(\nu\).
If \(F\) and \(G\) are the distribution functions corresponding to \(\mu\) and \(\nu\), respectively, then the distribution function corresponding to \(\mu\times \nu\) is given by \[ (F*G)(y)=(\mu*\nu)((-\infty,y])=\int_{-\infty}^\infty\nu((-\infty,y]-x)d\mu(x)\\ =\int_{-\infty}^\infty\nu((-\infty,y-x])d\mu(x)=\int_{-\infty}^\infty G(y-x)d\mu(x)=\int_{-\infty}^\infty G(y-x)dF(x)\\ \implies(F*G)(y)=\int_{-\infty}^\infty G(y-x)dF(x) \] * 정리하자면 convolution은 나중에 적분하는 변수에 대해 먼저 빼주고\((x)\) 적분한다\((y)\).