Nonnegative Combinations of Two Finite Measures, and the Average of Two Couplings

In the setting of Probability Measures on Euclidean Space and Random Vectors: Standing Notation, let $d\in\mathbb{N}$ satisfy $1\le d$.

1. (Nonnegative combinations of finite measures)(i) Let $(E,\mathcal{E})$ be a measurable space, let $\alpha$ and $\beta$ be measures on it with $\alpha(E)<\infty$ and $\beta(E)<\infty$, and let $s,t$ be nonnegative real numbers. Then the set function $\gamma$ on $\mathcal{E}$ given by

$$\gamma(A)=s\,\alpha(A)+t\,\beta(A)\qquad(A\in\mathcal{E})$$

is a measure on $(E,\mathcal{E})$ with $\gamma(E)<\infty$. Moreover, for every $f:E\to[0,\infty]$ measurable in the sense of Measurable Functions and the Lebesgue Integral: Standing Notation (iii),

$$\int_{E}f\,d\gamma=s\int_{E}f\,d\alpha+t\int_{E}f\,d\beta$$

in $[0,\infty]$; and every measurable $f:E\to\mathbb{R}$ that is integrable with respect to both $\alpha$ and $\beta$ is integrable with respect to $\gamma$ and satisfies the same identity, both sides being real.

2. (The average of two couplings)(ii) Let $\mu,\nu$ belong to the set $\mathcal{P}_{2}(\mathbb{R}^{d})$ of probability measures with finite second moment, let $\Pi(\mu,\nu)$ be the set of their couplings with quadratic cost $I$, and let $\pi,\pi'\in\Pi(\mu,\nu)$. Let $\hat{\pi}$ be the measure supplied by claim 1 for $E=\mathbb{R}^{d+d}$, $\mathcal{E}=\mathcal{B}(\mathbb{R}^{d+d})$, $\alpha=\pi$, $\beta=\pi'$ and $s=t=\tfrac{1}{2}$, so that $\hat{\pi}(A)=\tfrac{1}{2}\pi(A)+\tfrac{1}{2}\pi'(A)$ for every $A\in\mathcal{B}(\mathbb{R}^{d+d})$. Then $\hat{\pi}\in\Pi(\mu,\nu)$,

$$I(\hat{\pi})=\tfrac{1}{2}I(\pi)+\tfrac{1}{2}I(\pi'),$$

and $\pi(A)\le2\,\hat{\pi}(A)$ and $\pi'(A)\le2\,\hat{\pi}(A)$ for every $A\in\mathcal{B}(\mathbb{R}^{d+d})$. If moreover $\pi$ and $\pi'$ are optimal, then so is $\hat{\pi}$.