Let $X, Y \in \text{Random}(\mathbb{R}^D)$ each be a D4383: Random euclidean real number such that

(i)	$\cdot$ is the D743: Euclidean real dot product operation on $\mathbb{R}^D$

Then \begin{equation} \forall \, t \in \mathbb{R}^D : \mathbb{E}(e^{i t \cdot X}) = \mathbb{E}(e^{i t \cdot Y}) \quad \iff \quad X \overset{d}{=} Y \end{equation}

Let $X, Y \in \text{Random}(\mathbb{R}^D)$ each be a D4383: Random euclidean real number.

Then \begin{equation} \mathfrak{F}_X = \mathfrak{F}_Y \quad \iff \quad X \overset{d}{=} Y \end{equation}