Note on square root of density with value in Hilbert space

It does not include any clear assertion. I have no intention of proving any clear assertion in the future. Neveu, J., Bases mathematiques du calcul des probabilites, Deuxieme edition, Masson et Cie, Paris, 1970. Ay, N., Jost, J., Le, H. and Schwachhofer, L., Information geometry, Springer, Cham, 2017. [Note on the last sentence] In the sentence, $G$ was a complex Hilbert space and $\Omega$ was a measurable space. $\|f\|_Gfd\mu \in S^{\frac{1}{2}}(\Omega; G)$ is a $G$-valued measure on $\Omega$ as indefinite integral, but the operation of the indefinite integral is also the spectral measure on $\Omega$ of $S^{\frac{1}{2}}(\Omega; G)$ and $\|f\|_Gfd\mu$ also defines the finite measure $\|f\|_G^2d\mu$ on $\Omega$ as total variation. [Note] As a set, $S^{\frac{1}{2}}(\Omega; H)$ should probably be the set of all bounded-variation $H$-valued measures on $\Omega$. It seems possible to generalize to a measurable Hilbert bundle. In the case, such a characterization using vector measure seems difficult. Furthermore, suppose that a smooth bundle is given a smooth connection that is compatible with the metric. In the case, it seems that a flow on the base space determines a self-adjoint operator. [Problem] Let $\mu$ be a finite measure on $\Omega$. Although $L^2(\mu; H)\subset S^{\frac{1}{2}}(\Omega; L^2(\mu; H))$ holds, it would be more natural to see $L^2(\mu; H)\subset S^{\frac{1}{2}}(\Omega; H)$. The problem to be solved in the future is to define the concept of this kind of naturalness.