Convex Combinations of Probability Measures on Euclidean Space, and Finitely Supported Probability Measures as Combinations of 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 $n,M\in\mathbb{N}$, with finite sums $\sum_{i=1}^{M}$ and Dirac measures $\delta_{z}$ for $z\in\mathbb{R}^{n}$; finite subsets of $\mathbb{R}^{n}$ are Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §finite-sets.

1. (Convex combinations)¶ Let $\rho_{1},\dots,\rho_{M}\in\mathcal{P}(\mathbb{R}^{n})$ and let $c_{1},\dots,c_{M}\in\mathbb{R}$ be nonnegative with $\sum_{i=1}^{M}c_{i}=1$. Then the function

is a probability measure on $\mathbb{R}^{n}$. Let $f:\mathbb{R}^{n}\to[0,\infty]$ be Borel. If $\int_{\mathbb{R}^{n}}f\,d\rho_{i}$ is finite for every $i\in[M]$, then

if $\int_{\mathbb{R}^{n}}f\,d\rho_{i}=\infty$ for some $i\in[M]$ with $0<c_{i}$, then $\int_{\mathbb{R}^{n}}f\,d(\sum_{i=1}^{M}c_{i}\rho_{i})=\infty$. A Borel $f:\mathbb{R}^{n}\to\mathbb{R}$ that is integrable with respect to every $\rho_{i}$ is integrable with respect to $\sum_{i=1}^{M}c_{i}\rho_{i}$, with the same identity in $\mathbb{R}$.

2. (Finitely supported probability measures)¶ Let $z_{1},\dots,z_{M}\in\mathbb{R}^{n}$ be pairwise distinct, let $F=\{z_{1},\dots,z_{M}\}$, and let $\rho\in\mathcal{P}(\mathbb{R}^{n})$ satisfy $\rho(\mathbb{R}^{n}\setminus F)=0$. Then $\sum_{i=1}^{M}\rho(\{z_{i}\})=1$ and

the right side being the convex combination of clause 1.