Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that

(i)	$X, Y : \Omega \to \mathbb{R}$ are each a D3161: Random real number on $P$
(ii)	$X, Y$ is an D2713: Independent random collection on $P$
(iii)	$f_X$ and $f_Y$ are each a D209: Probability density function for $X$ and $Y$, respectively

Then $x \mapsto \int_{\mathbb{R}} f_X(x - y) \, f_Y(y) \, d y$ is a D209: Probability density function for $X + Y$.