My question is motivated by this link.
Let $(\Omega,\mathcal{F})$ and $(Y,\mathcal{B})$ be measurable spaces, a measurable map $T:\Omega\to Y$ is called a $Y$-valued random variable.
Now let $H$ be a separable real Hilbert space, $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space, and $Q:\Omega\to \mathcal{L}(X)$ be a measurable map taking values in the space of positive semidefinite, self-adjoint, trace class operator. One aim is to select measurable $\gamma_n:\Omega \to \mathbb{R}$ (eigenvalues) and $x_n:\Omega\to H$ (eigenvectors) such that $Q=\sum_{n=1}^\infty \gamma_n x_n\otimes x_n$.
By following the book by Zabczyk and Peszat, entitled "Stochastic Partial Differential Equations driven by Levy Noise" page 115, they apply the Kuratwoski-Ryll-Nardzesk section theorem, and select eigenvectors from the set $K:=\{x\in H: |x|_H\le 1\}$ endowed with the weak topology (so compact by Alagou).
My question is: does this procedure only prove the measurability of $x_n:\Omega\to H$, where $H$ is endowed with the weak topology? Is it equivalent to the original definition of being a random variable, whose topology is generated by the norm of $H$?
