demonstration).

Let us characterize the structure of ${\mathcal{M}}_r$. For any $\mu_{{LR}}\in {\mathcal{M}}_r$, by definition, it is supported on a $r$-dimensional linear subspace, whose orthonormal basis is denoted by $\left\{U_1, U_2, \cdots, U_r \right\}$. Then one could define a linear mapping ${{\boldsymbol{U}}}: {\mathbb{C}}^r\rightarrow {{\mathbb{C}}^{{n}}}$ by $${{\boldsymbol{U}}}: y\in {\mathbb{C}}^r \rightarrow \sum_{j=1}^{r} U_j y_j, \qquad \text{ or in matrix form } {{\boldsymbol{U}}}= \begin{bmatrix} U_1 & U_2 & \cdots & U_r \end{bmatrix}.$$ Let us denote the $r$-dimensional Stiefel manifold on ${{\mathbb{C}}^{{n}}}$ by $${V_r({{\mathbb{C}}^{{n}}})}:=  \left\{ {{\boldsymbol{U}}}: {\mathbb{C}}^r \rightarrow {{\mathbb{C}}^{{n}}}\text{ is linear, and }\ {{\boldsymbol{U}}}^{\dagger} {{\boldsymbol{U}}}= 
 {\mathrm{Id}}_{r\times r} \right\}.$$ Then, ${\mathcal{M}}_r$ could be viewed as a collection of measures in ${\mathcal{M}}$ with support on ${\text{Ran}}({{\boldsymbol{U}}})$ with bounded second moment, where ${\text{Ran}}({{\boldsymbol{U}}})$ is the range of some linear operator ${{\boldsymbol{U}}}\in {V_r({{\mathbb{C}}^{{n}}})}$. The restriction of $\mu_{{LR}}$ on $\text{Ran}({{\boldsymbol{U}}})$ can be represented as a finite signed measure on ${\mathbb{C}}^r$, given by the pullback $$\theta(E) := \mu_{{LR}}({{\boldsymbol{U}}}E) 
= \mu_{{LR}}(x \in {\text{Ran}}( {{\boldsymbol{U}}}): x\in {{\boldsymbol{U}}}E) = \mu_{{LR}}(x \in {\text{Ran}}( {{\boldsymbol{U}}}): {{\boldsymbol{U}}}^{\dagger} x\in E)$$ for any Borel set $E \subset {\mathbb{C}}^r$. Hence, for any measurable set $F \subset \text{Ran}({{\boldsymbol{U}}})\in \mathcal{B}$, $$\mu_{{LR}}(F) = \mu_{{LR}} ({{\boldsymbol{U}}}{{\boldsymbol{U}}}^{\dagger} F) = \theta( {{\boldsymbol{U}}}^{\dagger} F).$$ Thus, there exists a one-to-one correspondence between ${\mathcal{M}}_r$ and ${V_r({{\mathbb{C}}^{{n}}})}\oplus {{\mathcal{M}}_{{\mathbb{C}}^r}}$ where ${{\mathcal{M}}_{{\mathbb{C}}^r}}$ 