Dirac Measures on Euclidean Space: Probability Measure, Integrals, Push-Forwards and the Coupling of Two Dirac Measures

In the setting of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation, whose probability space $(\Omega,\mathcal{F},P)$ is not used, let $n\in\mathbb{N}$ and $x\in\mathbb{R}^{n}$, and let $\delta_{x}:\mathcal{B}(\mathbb{R}^{n})\to\mathbb{R}$ be the function with $\delta_{x}(B)=1$ if $x\in B$ and $\delta_{x}(B)=0$ if $x\notin B$. Probability measures and push-forwards are those of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward, Borel maps those of that clause, the concatenation $\iota^{n,n}$ that of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pairs, couplings $\Pi$ and the quadratic cost $I$ those of that definition, the second moment $M_{2}$ and $\mathcal{P}_{2}$ those of The Second Moment of a Probability Measure on Euclidean Space and the Probability Measures with Finite Second Moment §space, and $W_{2}$ the Wasserstein distance in every dimension. Then the following hold.

1. (Probability measure)¶ $\delta_{x}$ is a probability measure on $\mathbb{R}^{n}$; moreover $\delta_{x}\in\mathcal{P}_{2}(\mathbb{R}^{n})$ and $M_{2}(\delta_{x})=\lVert x\rVert^{2}$.

2. (Integrals)¶ For every Borel $f:\mathbb{R}^{n}\to[0,\infty]$, $\int_{\mathbb{R}^{n}}f\,d\delta_{x}=f(x)$. Every Borel $f:\mathbb{R}^{n}\to\mathbb{R}$ is integrable with respect to $\delta_{x}$, and $\int_{\mathbb{R}^{n}}f\,d\delta_{x}=f(x)$.

3. (Push-forwards)¶ For $m\in\mathbb{N}$ and Borel $h:\mathbb{R}^{n}\to\mathbb{R}^{m}$, $h_{\#}\delta_{x}=\delta_{h(x)}$, the right side being the function of the preamble formed in $\mathbb{R}^{m}$ at the point $h(x)$.

4. (Coupling of two Dirac measures)¶ For $x'\in\mathbb{R}^{n}$, the function $\delta_{\iota^{n,n}(x,x')}$ of the preamble formed in $\mathbb{R}^{n+n}$ satisfies $\delta_{\iota^{n,n}(x,x')}\in\Pi(\delta_{x},\delta_{x'})$ and $I(\delta_{\iota^{n,n}(x,x')})=\lVert x-x'\rVert^{2}$; consequently $W_{2}(\delta_{x},\delta_{x'})\le\lVert x-x'\rVert$.