Entropy of Products, Subadditivity over Two Marginals, and Convexity of the Entropy

In the setting of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation, let $q,p\in\mathbb{N}$. The entropy $\mathrm{Ent}$ and the set $\mathcal{P}_{2}^{\mathrm{Ent}}(\mathbb{R}^{m})$ of measures in $\mathcal{P}_{2}(\mathbb{R}^{m})$ with finite entropy are those of that definition, in each dimension $m$ used below; $\mu\boxtimes\nu$ and $\mathrm{pr}^{q,p}_{1},\mathrm{pr}^{q,p}_{2}$ are those of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pairs.

1. (Products)¶ For $\mu\in\mathcal{P}_{2}^{\mathrm{Ent}}(\mathbb{R}^{q})$ and $\nu\in\mathcal{P}_{2}^{\mathrm{Ent}}(\mathbb{R}^{p})$, $\mu\boxtimes\nu\in\mathcal{P}_{2}^{\mathrm{Ent}}(\mathbb{R}^{q+p})$ and $\mathrm{Ent}(\mu\boxtimes\nu)=\mathrm{Ent}(\mu)+\mathrm{Ent}(\nu)$.

2. (Subadditivity)¶ For $Q\in\mathcal{P}_{2}^{\mathrm{Ent}}(\mathbb{R}^{q+p})$, the marginals $(\mathrm{pr}^{q,p}_{1})_{\#}Q$ and $(\mathrm{pr}^{q,p}_{2})_{\#}Q$ belong to $\mathcal{P}_{2}^{\mathrm{Ent}}(\mathbb{R}^{q})$ and $\mathcal{P}_{2}^{\mathrm{Ent}}(\mathbb{R}^{p})$, and

3. (Convexity)¶ Let $n\in\mathbb{N}$, let $\mu_{1},\dots,\mu_{n}\in\mathcal{P}_{2}^{\mathrm{Ent}}(\mathbb{R}^{q})$, and let $t_{1},\dots,t_{n}\in\mathbb{R}$ be nonnegative with $\sum_{j=1}^{n}t_{j}=1$. The function $B\mapsto\sum_{j=1}^{n}t_{j}\mu_{j}(B)$ on $\mathcal{B}(\mathbb{R}^{q})$ is a probability measure $\bar\mu$, it belongs to $\mathcal{P}_{2}^{\mathrm{Ent}}(\mathbb{R}^{q})$, and $\mathrm{Ent}(\bar\mu)\le\sum_{j=1}^{n}t_{j}\,\mathrm{Ent}(\mu_{j})$.