Suppose that $X$ is a compact metric space, $\mathcal{H}$ is a Hilbert space, and that $A: \mathcal{B}(X) \rightarrow \mathcal{B}(\mathcal{H})$ is a Positive Operator Valued Measure (POVM), where $\mathcal{B}(X)$ denotes the Borel $\sigma$-algebra of subsets of $X$. That is,

- $A(\Delta)$ is a positive operator in $\mathcal{B}(\mathcal{H})$ for all $\Delta \in \mathcal{B}(X)$;
$A(\emptyset) = 0$ and $A(X) = \text{id}_{\mathcal{H}}$ (the identity operator on $\mathcal{H}$);

If $\{ \Delta_n \}_{n=1}^{\infty}$ is a sequence of pairwise disjoint sets in $\mathcal{B}(X)$, and if $g,h \in \mathcal{H}$, then

$$ \left \langle A\left( \bigcup_{n=1}^{\infty} \Delta_n \right)g , h \right \rangle = \sum_{n=1}^{\infty} \langle A(\Delta_n)g, h \rangle.$$

If $g,h \in \mathcal{H}$, consider the complex measure $A_{g,h}(\cdot) := \langle A(\cdot)g, h \rangle$. I am trying to show that the total variation norm of this measure is less than or equal to $||g||\cdot||h||$. This is true if $A$ is a projection valued measure, and I am pretty sure it is also true for POVM's, but I am not sure how to prove it for the POVM case. Any help or references would be appreciated! Thanks!