Suppose that we have the following data:
- $ \mathcal{G} $ is a locally compact Hausdorff groupoid, with its source and range maps denoted by $ s $ and $ r $ respectively.
- $ (\lambda^{x})_{x \in \mathcal{G}^{(0)}} $ is a left Haar system for $ \mathcal{G} $.
- $ \mu $ is a quasi-invariant measure on $ \mathcal{G}^{(0)} $.
- $ \mathfrak{H} \stackrel{\text{df}}{=} (\mathcal{H}_{x})_{x \in \mathcal{G}^{(0)}} $ is a measurable field of Hilbert spaces over $ \mathcal{G}^{(0)} $.
Now, let $ \nu $ denote the measure on $ \mathcal{G} $ induced by $ \mu $, i.e., $$ \forall E \in \mathscr{B}(\mathcal{G}): \qquad \nu(E) = \int_{\mathcal{G}^{(0)}} {\lambda^{x}}(E) ~ \mathrm{d}{\mu(x)}. $$ If $ \mathfrak{K} \stackrel{\text{df}}{=} (\mathcal{K}_{\gamma})_{\gamma \in \mathcal{G}} $ denotes the measurable field of Hilbert spaces over $ \mathcal{G} $ such that $ \mathcal{K}_{\gamma} = \mathcal{H}_{r(\gamma)} $ for each $ \gamma \in \mathcal{G} $, then is it necessarily true that the direct integrals $$ \int_{\mathcal{G}}^{\oplus} \mathcal{K}_{\gamma} ~ \mathrm{d}{\nu(\gamma)} \qquad \text{and} \qquad \int_{\mathcal{G}^{(0)}}^{\oplus} {L^{2}}(\mathcal{G}^{x},\mathcal{H}_{x}) ~ \mathrm{d}{\mu(x)} $$ are isomorphic as Hilbert spaces? I am asking this because for any $ \displaystyle f \in \int_{\mathcal{G}}^{\oplus} \mathcal{K}_{\gamma} ~ \mathrm{d}{\nu(\gamma)} $, we have \begin{align*} \int_{\mathcal{G}} \| f(\gamma) \|_{\mathcal{H}_{r(\gamma)}}^{2} ~ \mathrm{d}{\nu(\gamma)} & = \int_{\mathcal{G}^{(0)}} \left[ \int_{\mathcal{G}^{x}} \| f(\gamma) \|_{\mathcal{H}_{r(\gamma)}}^{2} ~ \mathrm{d}{{\lambda^{x}}(\gamma)} \right] \mathrm{d}{\mu(x)} \\ & = \int_{\mathcal{G}^{(0)}} \left\| f|_{\mathcal{G}^{x}} \right\|_{{L^{2}}(\mathcal{G}^{x},\mathcal{H}_{x})}^{2} ~ \mathrm{d}{\mu(x)}, \end{align*} so it appears that one can define a map $ \displaystyle T: \int_{\mathcal{G}}^{\oplus} \mathcal{K}_{\gamma} ~ \mathrm{d}{\nu(\gamma)} \to \int_{\mathcal{G}^{(0)}}^{\oplus} {L^{2}}(\mathcal{G}^{x},\mathcal{H}_{x}) ~ \mathrm{d}{\mu(x)} $ by $$ \forall f \in \int_{\mathcal{G}}^{\oplus} \mathcal{K}_{\gamma} ~ \mathrm{d}{\nu(\gamma)}, ~ \forall x \in \mathcal{G}^{(0)}: \qquad [T(f)](x) \stackrel{\text{df}}{=} f|_{\mathcal{G}^{x}}. $$ Thank you very much for your help!
