Bochner L2 space for a finite product-law input

Let $(\mathrm{\Gamma },\mathcal{B}(\mathrm{\Gamma }),\rho )$ be the finite product-law probability space of Finite product-law polynomial-chaos input data, and let $V$ be a separable real Hilbert space. Scalar integrals involving strongly measurable maps are taken using measurable versions: if simple maps converge to $U$ outside a measurable null set $N$, set $\tilde{U}=U$ off $N$ and $\tilde{U}=0$ on $N$, and set the approximants to zero on $N$. They then converge everywhere to $\tilde{U}$. For every open set $O\subseteq V$, convergence expresses ${\tilde{U}}^{-1}(O)$ through countably many eventual bounds on distances of the simple maps to $V\setminus O$, so this version is Borel measurable. Any two such versions agree outside a measurable null set. In particular, norm-square and pairing integrals below mean the integrals of $\mathrm{\parallel }\tilde{U}{\mathrm{\parallel }}_V^2$ and $(\tilde{U},\tilde{W}{)}_V$; their values are independent of the versions. This convention preserves arbitrary null-set values of strongly measurable maps and does not complete the underlying probability space.

Define $L_{\rho }^2(V)$ to be the space of strongly measurable maps $U:\mathrm{\Gamma }\to V$ for which ${\int }_{\mathrm{\Gamma }}\mathrm{\parallel }U(y){\mathrm{\parallel }}_V^{2}d\rho (y)<\mathrm{\infty }$, modulo equality $\rho$-almost everywhere. Its inner product is $(U,W{)}_{L_{\rho }^2(V)}={\int }_{\mathrm{\Gamma }}(U(y),W(y){)}_{V}d\rho (y)\mathrm{.}$