What does this notation (inner product with differential) mean?

Question

The following symbols are used in On optimal transport of matrix-valued measures by Yann Brenier and Dmitry Vorotnikov (cf. p. 3):

• $A : B := \text{tr}(A B^{\mathsf{T}}) = \sum_{i, j = 1}^{d} a_{i, j} b_{i, j}$ is the Frobenius inner product of two real $d \times d$ matrices $A, B \in \mathbb R^{d \times d}$.
• $\mathcal P^+$ is the subspace of real symmetric positive semidefinite $d \times d$ matrices.
• $\mathbb P^+$ is the set of $\mathcal P^+$-valued Radon measures $P$ on $\mathbb R^d$ with finite tr$(\text{d}P(\mathbb R^d))$.

Question 1. What is $\text{d} P$? Is $\text{d}P(\mathbb R^d) = P(\mathbb R^d) \in \mathcal P^+$? In the scalar case $d = 1$, I would expect $\text{d} P$ to be some sort of unnormalized density (with respect to the Lebesgue measure). All $\mathcal P^+$-valued measures $P$ have a density $P'$ with respect to the associated scalar trace measure $\text{tr}(P)$ (cf. p. 1-2 of Duran and Lopez-Rodriguez's The $L^p$ space of a positive definite matrix of measures and density of matrix polynomials in $L^1$), so is $\text{d}P$ maybe the density of $P$ with respect to the trace measure?

Question 2. I was wondering how to interpret (cf. p. 4) the real number (?) $$ \label{eq:star} \tag{$\star$} \int_{\mathbb R^d} \Phi : (\text{d} G_1 - \text{d} G_0) $$ for $G_0, G_1 \in \mathbb P^+$ and a bounded and Lipschitz continuous function $\Phi$ on $\mathbb R^d$ (probably mapping into $\mathbb R$, but this is not indicated), since the differential appears in the inner product. E.g. if $G_k = P_k \delta_{x_k}$ for $P_k \in \mathcal P^+$ and $x_k \in \mathbb R^d$ for $k \in \{ 0, 1 \}$, then I would suspect $G_0, G_1 \in \mathbb P^+$, but I don't know how to calculate \eqref{eq:star} for any $\Phi$.

If e.g. $G_0$ had a density $g_0$ with respect to some scalar measure $\lambda$, then I can imagine that $$\int_{\mathbb R^d} \Phi : \text{d}G_0 = \int_{\mathbb R^d} \Phi(x) : g_0(x) \; \text{d} \lambda(x).$$

I have encountered a similar notation (inner product / dual pairing with a differential) in the abstract of A simple proof of Singer's theorem by Hensgen and also in equation (7) of L. Ning and T. T. Georgiou. Metrics for matrix-valued measures via test functions.

I also searched for this notation in Diestel and Uhl's "Vector measures", bit didn't find it used.

Update

Since the entries $(G_0 - G_1)_{i, j}$ of $G_0 - G_1$ are scalar measures in their own right, can we interpret \eqref{eq:star} as the following sum of integrals of scalar functions against scalar measures? $$ \int_{\mathbb R^d} \sum_{i, j = 1}^{d} [\Phi(x)]_{i, j} \; \text{d}[(G_0 - G_1)_{i, j}](x) $$

Answer 1

Question 1: For any measure $\mu$, we often use the notation $\mathrm d\mu$ to mean the same thing as "$\mu$." In your context, $P$ (or $\mathrm dP$) is a matrix of measures, something like $$ P = \begin{pmatrix} \mu_{11} & \dots & \mu_{1d} \\ \vdots & \ddots & \vdots\\ \mu_{d1} & \dots & \mu_{dd} \end{pmatrix}, $$ so $P(\mathbb R^d)$ or $\mathrm dP(\mathbb R^d)$ (either of them) means the matrix of numbers $$ P(\mathbb R^d) = \begin{pmatrix} \mu_{11}(\mathbb R^d) & \dots & \mu_{1d}(\mathbb R^d) \\ \vdots & \ddots & \vdots\\ \mu_{d1}(\mathbb R^d) & \dots & \mu_{dd}(\mathbb R^d) \end{pmatrix}. $$

Question 2: For two matrices $\Phi,\Psi$, the notation $\Phi:\Psi$ means the same thing as $\mathrm{Tr}(\Phi\Psi^T)$ (this is implicitly defined at the top of p. 3).

So if $\Phi$ is a matrix-valued function and $\mu$ is a matrix-valued measure, to understand what $\int \Phi:\mathrm d\mu$ means, it is probably simplest to think about what it means when $\Phi$ is a simple function, i.e., takes only finitely many values. Say $\Phi = \sum_i A_i\mathbf 1_{E_i}$ for some matrices $A_i$ and some measurable sets $E_i\subset \mathbb R^d$. Then the only sensible meaning for the integral should be $$ \int \Phi:\mathrm d\mu = \sum_i A_i:\mu(E_i). $$ (Keep in mind how $\mu(E_i)$ is a matrix of numbers for each $i$, so we can fall back on the notation $\Phi:\Psi$ when $\Phi,\Psi$ are both matrices of numbers.) To generalize to integrable functions $\Phi$ should now be a routine application of the standard machinery of Lebesgue integration.

Lastly, if the preceding is understood, $\int \Phi:(\mathrm d\mu_1 - \mathrm d\mu_2)$ should be understood in the analogous way as usual integration against a signed measure.