One might naively think that a collection (left(X_{t}, t in mathbb{R}^{+} ight))of independent r.v's may be chosen

One might naively think that a collection \(\left(X_{t}, t \in \mathbb{R}^{+}\right)\)of independent r.v's may be chosen "measurably," i.e., with the map

\[\left(\mathbb{R}^{+} \times \Omega, \mathcal{B}_{\mathbb{R}^{+}} \times \mathcal{F}\right) \rightarrow\left(\mathbb{R}, \mathcal{B}_{\mathbb{R}}\right):(t, \omega) \rightarrow X_{t}(\omega)\]

being measurable, so that \(X\) is a "true" process. Prove that if the \(X_{t}\) 's are centered and \(\sup _{t} \mathbb{E}\left(X_{t}^{2}\right)<\infty\), then no measurable choice can be constructed, except \(X=0\).