I have to show that for a family of maps $X = \{X_i\}_{i \in I}$ from a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ into a measure space $(E, \mathcal{E})$, where $\mathcal{E}^I$ is the product sigma algebra of $E^I$ the following holds:
$X: \Omega \rightarrow E^I$ is $\mathcal{F}-\mathcal{E}^I$ measurable $\iff$ $\forall i \in I: X_i: \Omega \rightarrow E$ is $\mathcal{F}-\mathcal{E}$ measurable.
I have trouble proving the direction $\Leftarrow$ and doubt that it holds. If it does hold, can anybody point me into the right direction?