Assume that $(\Omega,\mathcal{F},\mathbb{P})$ is a probability space, and $E$ a separable Banach space, and let $f \colon \Omega \to E$ be a random variable with values in $E$. This implies that $f$ be measurable from $(\Omega,\mathcal{F})$ to $(E,\mathcal{B}(E))$, where $\mathcal{B}(E)$ is the Borel $\sigma$ field associated with $E$.
If $V \subset E$ is a subspace of $E$, under which conditions is $f$ also a $V$-valued random variable?
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