\infty$$ for each $\phi \in C_0(X)$ and $\phi \in C_b(X)$, respectively.

Whenever $X=\mathbb{R}^d$, $d\ge 1$, it remains understood that it is endowed with the Euclidean norm (and induced distance), which shall be simply denoted by $|\cdot|$.

For a Lipschitz function $f\colon X\to\mathbb{R}$ we denote by $$\mathrm{Lip}(f)\coloneqq\sup_{x,y\in X \atop x\neq y}\frac{|f(x)-f(y)|}{{\mathsf d}_X(x,y)}$$ the Lipschitz constant. The notations $\mathrm{Lip}(X)$ and $\mathrm{Lip}_b(X)$ will be used for the spaces of Lipschitz and bounded Lipschitz functions on $X$, respectively. Both are normed spaces with the norm $\lVert f\rVert \coloneqq \lVert f\rVert_\infty+ \mathrm{Lip}(f)$.

In a complete and separable metric space $(X,{\mathsf d}_X)$, we shall use the Kantorovich-Rubinstein distance $\mathcal{W}_1 $ in the class ${\mathcal P}(X)$, defined as $$\mathcal{W}_1(\mu,\nu)\coloneqq\sup\bigg\{\int_X\varphi\,\mathrm{d}\mu-\int_X\varphi\,\mathrm{d}\nu: \varphi\in\mathrm{Lip}_b(X), \,\mathrm{Lip}(\varphi){\leqslant}1\bigg\}$$ or equivalently (thanks to the Kantorovich duality) as $$\mathcal{W}_1(\mu,\nu)\coloneqq\inf\bigg\{\int_{X\times X} {{}{\mathsf d}_X(x,y)}\,\mathrm{d}\Pi(x,y): 
\Pi(A\times X)=\mu(A),\,\,\Pi(X\times B)=\nu(B)\bigg\},$$ involving couplings $\Pi$ of $\mu$ and $\nu$. Notice that ${\mathcal W}_1(\mu,\nu)$ is finite if $\mu$ and $\nu$ belong to the space $${\mathcal P}_1(X)\coloneqq \bigg\{\mu\in{\mathcal P}(X):  \text{$\int_X {\mathsf d}_X(x,\bar x)\,\mathrm{d}\mu(x)<+\infty$ for some $\bar x\in X$}\bigg\}$$ and that $({\mathcal P}_1(X),\mathcal{W}_1)$ is complete if $(X,{\mathsf d}_X)$ is complete. For a positive measure $\mu \in\mathcal{M}_+(E)$, for $E$ being a Banach space, we define the first moment $m_1(\mu)$ as $$m_1(\mu)\coloneqq\int_{E} \lVert x \rVert_E\,\mathrm{d}\mu\,.$$ Notice that, for 