In Kallenberg's Foundations of Modern Probability, he defines a $\sigma$-field $\mathscr{F}$ is $\Bbb P$-trivial if $\Bbb PA = 0$ or $1$ for every $A \in \mathscr{F}$. We say a random element $\xi$ taking value in a measurable space $(S,\mathcal{S})$ is a.s. degenerate or a.s. constant if there exists $s \in S$ such that $\Bbb P \{\xi = s\} = 1$. Kallenberg proves the following:
Lemma: Let $\mathscr{F}$ be $\Bbb P$-trivial and $\xi : \Omega \to S$ be $\mathscr{F}$-measurable. Then if $S$ is a seperable metric space with $\mathcal{S}$ the corresponding Borel $\sigma$-field, $\xi$ is a.s. constant.
Does this Lemma hold if $S$ is not assumed to take values in a separable metric space? Assuming the Lemma fails without this hypothesis, what is a counterexample.