Let $X$ be a locally compact Polish space. Consider the set $M(X; S^d)$ of real symmetric matrix-valued measures on $X$, that is, the set of countably (with respect to the Frobenius norm $\| \cdot \|_F$) additive functions $u \colon B(X) \to S^d$ with $u(\emptyset) = 0$ (and tr$(u(X)) < \infty$), where $B(X)$ is the Borel $\sigma$-algebra of $X$. The total variation measure $| u |$ associated to $u \in M(X; S^d)$ is defined by $$ | u |(E) = \sup\left\{ \sum_{k \in \mathbb N} \| u(A_k) \|_{F}: (A_k)_{k \in \mathbb N} \subset B(X) \text{ is a pairwise disjoint partition of } E \right\} $$ for $E \in B(X)$. For a spike train, a linear combination of Dirac measures, $$ u = \sum_{k = 1}^{N} P_k \delta_{\{ x_k \}}, $$ where $(P_k)_{k = 1}^{N} \subset S^d$ and $(x_k)_{k = 1}^{N} \subset X$, we have $$ | u | = \sum_{k = 1}^{N} \| P_k \|_F \delta_{x_k}. $$
My question
I am wondering if there is an explicit form for $\| u \|$ for non-discrete measures (perhaps we need to choose another norm than $\| \cdot \|_F$, e.g. the nuclear norm $\text{tr}(| \cdot |)$ or the spectral norm $\| \cdot \|_2$?). It would be great if we can somehow relate $| u |$ to the associated trace measure $\text{tr}(u)$.
Thoughts. Since the trace norm $A \mapsto \text{tr}(| A |)$ reduces to the trace $A \mapsto \text{tr}(A)$ for $A \in S_+^d$, is the trace measure just the total variation measure if the norm is the trace norm instead of the Frobenius norm?
I was thinking that we can decompose a symmetric matrix $P$ into the the difference of two positive (semi)definite matrices $P_1 - P_2$, so maybe $\| P \| = \| P_1 \| + \| P_2 \|$ for such a decomposition?
Another thought is that the entries $u_{i, j}$ of a real symmetric matrix-valued measure are real-valued measures in their one right, with total variation measure $| u_{i, j} | = u_{i, j}^+ + u_{i, j}^-$ (Hahn-Jordan decomposition). Is there any relationship between the $| u_{i, j} |$ and $\| u \|$?
The motivation for this is that integrals of functions with respect to $\text{tr}(u)$ are much easier to handle that with respect to the abstract non-closed form $\| u \|$. For measures $u$ valued only in the positive semidefinite matrices $S_+^d$, we have that they are absolutely continuous with respect to $\text{tr}(u)$, but this is not the case for symmetric-valued matrix measures. For $P \in S^d$ we have the inequality $\| P \|_2 \le \| P \|_F \le \text{tr}(P)$.
We also have the following relationship between the trace measure and the total variation measure: the pre-dual of $M(X; S^d)$ are the continuous functions on $X$ with values in $S^d$ that vanish at infinity, $C_0(X; S^d)$ via the duality pairing (see Singer's theorem) $$ \langle u, f \rangle_{M(X; S^d) \times C_0(X; S^d)} := \int_{X} \langle f(x), u'(x) \rangle_{F} \, \text{d}| u |(x) $$ where $u \in M(X; S^d)$, $f \in C_0(X; S^d)$ and $u'$ is the Radon-Nikodym derivative of $u$ with respect to $| u |$. On the other hand, we have, due to the linearity of the integral, $$ \langle u, f \rangle_{M(X; S^d) \times C_0(X; S^d)} = \text{tr}\left( \int_{X} f(x) \, \text{d}u(x) \right) = \int_{X} \langle f(x), u_{\text{tr}}'(x) \rangle \, \text{d}[\text{tr}(u)](x), $$ where $u_{\text{tr}}'$ is the Radon-Nikodym derivative of $u$ with respect to $\text{tr}(u)$.
Update
Thanks so @Adam's comment, we must only consider positive matrix measures $u \in M(X; S_+^d)$, because any $u \in M(X, S^d)$ can be pointwise decomposed as $u = O \otimes P$, where $O$ is orthogonal-matrix valued and $P \in M(X; S_+^d)$. Since the three matrix norms proposed above are al unitarily invariant, we have $| u | = | P |$.