The Direct Sum of Hilbert Spaces

Let $\left\{ (X_i, \Omega_i, \mu_i) : i \in I\right\}$ be a collection of measure spaces and define $X$, $\Omega$, and $\mu$ as follows. Let $X =$ the disjoint union of $\left\{ X_i : i \in I\right\}$ and let $\Omega = \left\{ \Delta \subseteq X : \Delta \cap X_i \in \Omega_i \text{ for all } i\right\}$. For $\Delta$ in $\Omega$ put $\mu (\Delta ) = \sum_i \mu_i(\Delta \cap X_i)$. Show that $(X, \Omega , \mu )$ is a measure space and $L^2(X, \Omega , \mu )$ is isomorphic to $\bigoplus \left\{ L^2(X_i, \Omega_i, \mu_i) : i \in I\right\}$.

Let $(X, \Omega )$ be a measurable space, let $\mu_1, \mu_2$ be $\sigma$-finite measures defined on $(X, \Omega )$, and put $\mu = \mu_1 + \mu_2$. Show that the map $V : L^2(X, \Omega , \mu ) \to L^2(X, \Omega , \mu_1) \oplus L^2(X, \Omega , \mu_2)$ defined by $Vf = f_1 \oplus f_2$, where $f_j$ is the equivalence class of $L^2(X, \Omega , \mu_j)$ corresponding to $f$, is well defined, linear, and injective. Show that $V$ is an isomorphism iff $\mu_1$ and $\mu_2$ are mutually singular.