[Comment 0] : Let $\pi_1, \ldots, \pi_N$ each be a D327: Canonical set projection on $\mathbb{R}^N$ such that \begin{equation} \pi_n : \mathbb{R}^N \to \mathbb{R}, \quad \pi_n(x_1, \ldots, x_n \ldots, x_N) = x_n \end{equation} for each $n \in 1, \ldots, N$. If $E \in \mathcal{B}(\mathbb{R}^N)$, then result R4602: Element in finite cartesian product iff components in images of canonical projections shows that $X \in E$ if and only if \begin{equation} X_1 \in \pi_1 E, \quad \ldots, \quad X_N \in \pi_N E \end{equation} Thus we can write the condition also as \begin{equation} \forall \, E \in \mathcal{B}(\mathbb{R}^N) : \mathbb{P}(X_1 \in \pi_1 E, \ldots, X_N \in \pi_N E) = \int_E f(x) \, \upsilon(d x) \end{equation}