Let $X, Y \in \mathsf{Random}(\mathbb{R}^D)$ each be a D4383: Random euclidean real number.
Let $\mu_X, \mu_Y : \mathcal{B}(\mathbb{R}^D) \to [0, \infty]$ each be a D5241: Standard borel unsigned basic measure on $\mathbb{R}^D$ such that

(i)	\begin{equation} \mu_X(\mathbb{R}^D), \mu_Y(\mathbb{R}^D) < \infty \end{equation}
(ii)	\begin{equation} \forall \, E \in \mathcal{B}(\mathbb{R}^D) : \mathbb{P}(X \in E) = \mu_X(E) \end{equation}
(iii)	\begin{equation} \forall \, E \in \mathcal{B}(\mathbb{R}^D) : \mathbb{P}(Y \in E) = \mu_Y(E) \end{equation}

Let $B \in \mathcal{B}(\mathbb{R}^D)$ be a D5112: Standard Euclidean real Borel set.

Then \begin{equation} \mathbb{P} (X + Y \in B) = (\mu_X * \mu_Y)(B) = \int_{\mathbb{R}^D} \int_{\mathbb{R}^D} I_B(x + y) \, \mu_X(d x) \, \mu_Y(d y) \end{equation}